Unique input-response relationships from unambiguous parameter sets

ABSTRACT

Non-mechanistic, differential-equation-free approaches for unambiguously predicting a particular structure-activity responses of a system to a given molecular structure input are provided in the form of systems, methods, and devices. These approaches provide one-to-one relationships between model parameters and model output to obtain a specificity of results between model parameters and model output to get a unique input-response relationship. The systems, methods, and devices (i) reduce the cost of research and development by offering an accurate modeling of heterogeneous and complex physical systems; (ii) reduce the cost of creating such systems and methods by simplifying the modeling process; (iii) accurately capture and model inherent nonlinearities in cases where sufficient knowledge does not exist to a priori build a model and its parameters; and, (iv) provide one-to-one relationships between model parameters and model outputs, addressing the problem of the ambiguities inherent in the current, state-of-the-art systems and methods.

CROSS REFERENCE TO RELATED CASES

This application is a continuation of U.S. patent application Ser. No.15/990,646, filed May 27, 2018, which is a continuation U.S. patentapplication Ser. No. 15/254,514, filed Sep. 1, 2016, now U.S. Pat. No.10,013,515, which is a continuation of U.S. patent application Ser. No.13/962,961, filed Aug. 9, 2013, now U.S. Pat. No. 9,460,246, which is acontinuation of U.S. patent application Ser. No. 13/717,644, filed Dec.17, 2012, now U.S. Pat. No. 8,554,712, each of which is herebyincorporated herein by reference in its entirety.

BACKGROUND Field of the Invention

The teachings generally relate to a non-mechanistic,differential-equation-free approach for predicting a particularstructure-activity responses of a mammalian system to a given molecularstructure input.

Description of the Related Art

Research and development has historically relied on physical modeling todevelop new technologies. Given the speed at which computers can performcomputations, and the vast amount of computer memory available, computermodeling allows us to speed-up and reduce costs of research byfacilitating the creation of a large number of simulations over a widerange of physical scales very quickly. As with physical modeling,computer modeling and simulation deals with first characterizing andthen predicting input-response type relationships. What type of reactionwill occur between two chemicals? What is the flow response when a givenamount of water is introduced into a particular porous media? How willthe components of a watershed—rivers, reservoirs, aquifers, etc.—reactwhen subjected to a given rainfall or contamination event? How will aperson's blood glucose level respond to a given meal? How will adiseased tissue respond to a drug regimen? These are allinput-response-type questions that can be addressed throughmathematical/computational modeling and simulation. Generally speaking,this can be referred to as “input-response modeling”. In the field ofdrug design, this can also be referred to as “dose-response modeling.”An accurate model will give researchers a way of running simulations toquickly observe and test a large number of complex input-responsephenomena that might be too costly and time-consuming to observe andtest in a real-world setting.

The reliance on physical modeling can be very expensive, which makes theuse of computer modeling an attractive way to reduce costs. For example,the average drug developed by a major pharmaceutical company costs atleast $4 billion, and it can be as much as $11 billion. The range ofmoney spent is quite wide, for example, as AstraZeneca has spent about$12 billion in research money for every new drug approved; Eli Lillyspent about $4.5 billion per drug; and, Amgen has spent about $3.7billion per drug. The costs are so high, at least in part, becausesingle clinical trial can cost $100 million, and the combined cost ofmanufacturing and clinical testing for some drugs can add up to $1billion. Computer modeling of drugs, if improved such that it can bedone efficiently and effectively, can cut costs and help make thebusiness of drug discovery more attractive. Other industries, of course,can also benefit from such efficient and effective computer modelingmethods.

State-of-the-art systems and methods, however, typically use mechanisticcomputer models to try and avoid the costs of physical modeling.Unfortunately, such models can be very complex, insufficient andambiguous, and moreover, lacking in accuracy. Such models useestablished empirical formulas as “first principles” that provide theframework to make “mechanistic” predictions. Complex biological systemscan be modeled, for example, using laboratory experiments to establishsuch first-principle-type relationships between components of thesystem. For example, laboratory experiments can be used to determine theways in which a certain disease progresses in the human body, and thiscan be used to help predict how effective a drug might be in stopping,or slowing down, the progression of a disease.

Unfortunately, the current, state-of-the-art approaches have someserious limitations. There are problems, for example, in dealing withheterogeneous and complex systems, in that the models fail byinsufficiently characterizing the systems. Predicting the flow ofrainfall through the ground to an adjacent stream, for example, involvesa complex and heterogeneous combination of media types in the ground.The variations throughout the media make it difficult-to-impossible toapply Darcy's Law accurately in such a complex system. And, althoughpossible in theory, accurately identifying and modeling such complex andheterogeneous media throughout the system is often considered costprohibitive, as well as time prohibitive in many cases. As the systemsbecome more mechanistically complex, of course, we need more empiricalrelationships and a more complex model. Hydraulic conductivitymechanisms may not be enough, for example, as there can also be chemicalreaction mechanisms affecting the movement of the fluids. Humanbiological systems are examples of highly complex systems that aredifficult to scale from the lab to the human body, as measurements thatcan be taken in the lab may not be obtainable in the human body, forexample. In predicting the response of a tumor to a drug, for example,measuring in vitro or ex vivo tumor size and growth in small time scalesis one thing, but getting such in vivo measurements can bedifficult-to-impossible. In addition, a system may have nonlinearitiesthat need to be addressed, requiring further and often futile attemptsat adjusting the mechanistic model. Moreover, current models oftencannot map input properties to model parameters. This is because theylack the necessary one-to-one relationships between model parameters andmodel output. This lack of specificity results in an ambiguity betweenmodel parameters and output that makes it impossible to get uniqueinput-response relationships, such that the same input can produce awide range of responses, or many different inputs could produce the sameresponse.

Accordingly, one of skill will appreciate a data-based, non-mechanistic,differential-equation-free approach for predicting a particular responseof a system to a given input. In particular, one of skill willappreciate having the ability to (i) reduce the cost of research anddevelopment by offering an accurate modeling of heterogeneous andcomplex physical systems; (ii) reduce the cost of creating such systemsand methods by simplifying the modeling process; (iii) accuratelycapture and model inherent nonlinearities in cases where sufficientknowledge does not exist to a priori build a model and its parameters;and, (iv) provide one-to-one relationships between model parameters andmodel outputs, addressing the problem of the ambiguities inherent in thecurrent, state-of-the-art systems and methods.

SUMMARY

The teachings generally relate to a non-mechanistic,differential-equation-free approach for predicting a particular responseof a system to a given input. In some embodiments, the teachings aregenerally directed to a non-compartmental method of predicting atime-dependent response of a component of a system to an input into thesystem. The method can comprise identifying the system, the component,the input, and the time-dependent response; wherein, the input includesa set of actual inputs and a test input, and the time-dependent responseincludes a set of time-dependent actual responses and a test response;obtaining the set of time-dependent actual responses of the component tothe set of actual inputs; and, using the set of actual inputs and theset of time-dependent actual responses to provide a model for predictingthe test response to the test input, the model comprising the formula:

$\begin{matrix}{{{C(t)}\lbrack {M_{0}^{0} + {M_{0}^{1}({kernel})}} \rbrack} + {\quad{{\lbrack {M_{1}^{0} + {M_{1}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}{1 + {( {e^{K} - 2} )e^{{- {\lbrack{N_{1}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}} \}} + \ldots\mspace{11mu} + {\lbrack {M_{n}^{0} + {M_{n}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}{1 + {( {e^{K} - 2} )e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}} \}}}}} & (9)\end{matrix}$

-   -   wherein,    -   M⁰ ₀, . . . , M⁰ _(n) and M¹ ₀, . . . , M¹ _(n) are overall        scaling parameters;    -   N⁰ ₁, . . . , N⁰ _(n) and N¹ ₁, . . . , N¹ _(n) are exponential        scaling parameters;    -   n ranges from 1 to 4;    -   K is an overall shifting parameter; and,    -   C(t) is the time-dependent response to the test input at time t;    -   and,

${{kernel} \equiv \frac{1 - e^{{- \alpha_{p}}C_{0}}}{1 + {( {e^{K_{p}} - 2} )e^{{- \alpha_{p}}C_{0}}}}};$

-   -   wherein, C₀ is the initial amount of the input; K_(p) is a        shifting parameter related to C₀; and, α_(p) is shifting and        scaling parameter related to C₀.

The teachings include a non-compartmental method of predicting atime-dependent response of a component of a mammalian system to an inputinto the system. And, it should be appreciated that the response can bemeasured in vivo, in vitro, or ex vivo, in some embodiments. The methodscan also comprise selecting a component of the system, the componentselected from the group consisting of a cell, a tissue, an organ, a DNA,a virus, a protein, an antibody, a bacteria; selecting a set of actualinputs, the set of actual inputs having an element selected from thegroup consisting of a DNA, a virus, a protein, an antibody, a bacteria,a chemical, a dietary supplement, a nutrient, and a drug; obtaining aset of time-dependent actual responses of the component to the set ofactual inputs; and, using the set of actual inputs and the set oftime-dependent actual responses to provide a model for predicting a testresponse to a test input, the model comprising the formula

$\begin{matrix}{{{C(t)}\lbrack {M_{0}^{0} + {M_{0}^{1}({kernel})}} \rbrack} + {\quad{{\lbrack {M_{1}^{0} + {M_{1}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}{1 + {( {e^{K} - 2} )e^{{- {\lbrack{N_{1}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}} \}} + \ldots\mspace{11mu} + {\lbrack {M_{n}^{0} + {M_{n}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}{1 + {( {e^{K} - 2} )e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}} \}}}}} & (9)\end{matrix}$

-   -   wherein,    -   M⁰ ₀, . . . , M⁰ _(n) and M¹ ₀, . . . , M¹ _(n) are overall        scaling parameters;    -   N⁰ ₁, . . . , N⁰ _(n) and N¹ ₁, . . . , N¹ _(n) are exponential        scaling parameters;    -   n ranges from 1 to 4;    -   K is an overall shifting parameter; and,    -   C(t) is the time-dependent response to the test input at time t;    -   and,

${{kernel} \equiv \frac{1 - e^{{- \alpha_{p}}C_{0}}}{1 + {( {e^{K_{p}} - 2} )e^{{- \alpha_{p}}C_{0}}}}};$

-   -   wherein, C₀ is the initial amount of the test input; K_(p) is a        shifting parameter related to C₀; and, α_(p) is shifting and        scaling parameter related to C₀.

In some embodiments, the teachings are directed to a device forpredicting a time-dependent response of a component of a physical systemto an input into the system. In these embodiments, the device cancomprise a processor; a database for storing a set of actual input data,a set of time-dependent actual response data, test input data, andtime-dependent test response data on a non-transitory computer readablemedium; an enumeration engine on a non-transitory computer readablemedium to parameterize a non-compartmental model for predicting a testresponse to a test input, the non-compartmental model comprising theformula

$\begin{matrix}{{{C(t)}\lbrack {M_{0}^{0} + {M_{0}^{1}({kernel})}} \rbrack} + {\quad{{\lbrack {M_{1}^{0} + {M_{1}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}{1 + {( {e^{K} - 2} )e^{{- {\lbrack{N_{1}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}} \}} + \ldots\mspace{11mu} + {\lbrack {M_{n}^{0} + {M_{n}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}{1 + {( {e^{K} - 2} )e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}} \}}}}} & (9)\end{matrix}$

-   -   wherein,    -   M⁰ ₀, . . . , M⁰ _(n) and M¹ ₀, . . . , M¹ _(n) are overall        scaling parameters;    -   N⁰ ₁, . . . , N⁰ _(n) and N¹ ₁, . . . , N¹ _(n) are exponential        scaling parameters;    -   n ranges from 1 to 4;    -   K is an overall shifting parameter; and,    -   C(t) is the time-dependent response to the test input at time t;    -   and,

${{kernel} \equiv \frac{1 - e^{{- \alpha_{p}}C_{0}}}{1 + {( {e^{K_{p}} - 2} )e^{{- \alpha_{p}}C_{0}}}}};$

-   -   wherein, C₀ is the initial amount of the test input; K_(p) is a        shifting parameter related to C₀; and, α_(p) is shifting and        scaling parameter related to C₀;        and, a transformation module on a non-transitory computer        readable medium to transform the test data into the        time-dependent response data using the non-compartmental model.

The systems can be virtually any physical or non-physical system knownto one of skill in which that person of skill may want to predict aparticular response of the system to a given input. In some embodiments,the system can be an environmental system, and the component can beselected from the group consisting of air, water, and soil. In someembodiments, the system can be a mammal, and the component can beselected from the group consisting of a cell, a tissue, an organ, a DNA,a virus, a protein, an antibody, a bacteria. In some embodiments, thesystem can be a chemical system, a biological system, a mechanicalsystem, an electrical system, a financial system, a sociological system,a political system, or a combination thereof. As such, the teachingsprovided herein include general methods of predicting a particularresponse of any such system to a given input. For example, a biologicalsystem can have a biological input, a mechanical system can have amechanical data input, an electrical system can have a relativeelectrical data input, a financial system can have a relative financialdata input, a sociological system can have a relative sociological datainput, a political system can have a relative political data input, andthe like.

In some embodiments, the teachings are directed to a device forpredicting a time-dependent response of a component of a mammaliansystem to an input into the system. In these embodiments, the device cancomprise a processor; a database for storing a set of actual input data,a set of time-dependent actual response data, test input data, andtime-dependent test response data on a non-transitory computer readablemedium; an enumeration engine on a non-transitory computer readablemedium to parameterize a non-compartmental model for predicting a testresponse to a test input, the non-compartmental model comprising theformula

$\begin{matrix}{{{C(t)}\lbrack {M_{0}^{0} + {M_{0}^{1}({kernel})}} \rbrack} + {\quad{{\lbrack {M_{1}^{0} + {M_{1}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}{1 + {( {e^{K} - 2} )e^{{- {\lbrack{N_{1}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}} \}} + \ldots\mspace{11mu} + {\lbrack {M_{n}^{0} + {M_{n}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}{1 + {( {e^{K} - 2} )e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}} \}}}}} & (9)\end{matrix}$

-   -   wherein,    -   M⁰ ₀, . . . , M⁰ _(n) and M¹ ₀, . . . , M¹ _(n) are overall        scaling parameters;    -   N⁰ ₁, . . . , N⁰ _(n) and N¹ ₁, . . . , N¹ _(n) are exponential        scaling parameters;    -   n ranges from 1 to 4;    -   K is an overall shifting parameter; and,    -   C(t) is the time-dependent response to the test input at time t;    -   and,

${{kernel} \equiv \frac{1 - e^{{- \alpha_{p}}C_{0}}}{1 + {( {e^{K_{p}} - 2} )e^{{- \alpha_{p}}C_{0}}}}};$

-   -   wherein, C₀ is the initial amount of the test input; K_(p) is a        shifting parameter related to C₀; and, α_(p) is shifting and        scaling parameter related to C₀;        and, a transformation module on a non-transitory computer        readable medium to transform the test data into the        time-dependent response data using the non-compartmental model.

Any desired component known to one of skill can be used, in which thedesired component is a component of interest to the person of skill. Insome embodiments, the component can be blood, a tumor cell, a virus, abacteria, or a combination thereof.

Any desired test response known to one of skill can be used, in whichthe desired test response is a response of interest to the person ofskill. In some embodiments, the test response is a bacterial load, aviral load, a tumor marker, a blood chemistry, or a combination thereof.

Any desired set of actual inputs known to one of skill can be used, inwhich the desired set of actual inputs are of interest to the person ofskill. In some embodiments, the set of actual inputs can include a setof dosages of a drug, a set of drugs, or a combination thereof.

Any desired input known to one of skill can be used, in which thedesired input is of interest to the person of skill. In someembodiments, the input is a diabetes drug, and the time-dependentresponse can be glucose in the bloodstream.

The systems, methods, and devices taught herein transform input datainto response data and, as such, can be used to obtain thetime-dependent test response to the test input. And, the devices taughtherein can be in any form, whether handheld, desktop, intranet,internet, or otherwise cloud-based. In some embodiments, the device canbe a handheld device including, but not limited to, a PDA, a smartphone,an iPAD, a personal computer, and the like, including devices that arenot intended for any other substantial use.

BRIEF DESCRIPTION OF THE DRAWINGS

FIG. 1 shows a general technology platform for systems that can be usedin the practice of the methods taught herein, according to someembodiments.

FIG. 2 illustrates a processor-memory diagram to describe components ofa system, according to some embodiments.

FIG. 3 is a concept diagram illustrating a system taught herein,according to some embodiments.

FIG. 4 shows an example of a prior art, simple two-compartment linearmodel, with forward (k_(f)) and reverse (k_(r)) reactions between thetwo compartments as well as elimination (k_(e)) from the secondcompartment, according to some embodiments.

FIG. 5 illustrates a flowchart for a non-compartmental method ofpredicting a time-dependent response of a component of a system to aninput into the system, according to some embodiments.

FIG. 6 shows how a network may be used for the systems and methodstaught herein, in some embodiments.

FIG. 7 shows a prior art, two-compartment linear model that wasconstructed to model the PK behavior of a particular drug, according tosome embodiments.

FIG. 8 shows the data used to calibrate this model (find optimalparameter values) a two-compartment linear model that was constructed tomodel the PK behavior of a particular drug, according to someembodiments.

FIG. 9 shows a linear two-compartment model solute on for C_(p)(t)compared to data for the pharmacokinetic modeling, according to someembodiments.

FIG. 10 shows the C_(p)(t) response function compared to the data foreach of the 25 mg, 100 mg, and 400 mg cases, according to someembodiments.

FIG. 11 shows the P(t) response function compared to data for the enzymereaction modeling, according to some embodiments.

FIGS. 12A and 12B illustrate the pharmacokinetic and pharmacodynamicmodel as used in predicting viral loads in response to administration oftenofovir, according to some embodiments.

FIG. 13 shows a plot of the responses provided using the systems andmethods taught herein as compared to the large-scale compartment model,according to some embodiments.

FIG. 14 shows a three-compartment model that is used as a simplerepresentation for the absorption of a compound between the intestinesand bloodstream for a dosing study, according to some embodiments.

FIG. 15 shows the prediction of the bloodstream concentration vs. timeprofile for a 1000 mg dose, using both the linear and systems andmethods taught herein, according to some embodiments.

DETAILED DESCRIPTION

Non-mechanistic, differential-equation-free approaches for predicting aparticular response of a system to a given input are provided in theform of systems, methods, and devices. These approaches are generallydirected to a non-compartmental method of predicting a time-dependentresponse of a component of a system to an input into the system. Thesystems, methods, and devices provide the ability to (i) reduce the costof research and development by offering an accurate modeling ofheterogeneous and complex physical systems; (ii) reduce the cost ofcreating such systems and methods by simplifying the modeling process;(iii) accurately capture and model inherent nonlinearities in caseswhere sufficient knowledge does not exist to a priori build a model andits parameters; and, (iv) provide one-to-one relationships between modelparameters and model outputs, addressing the problem of the ambiguitiesinherent in the current, state-of-the-art systems and methods.

FIG. 1 shows a general technology platform for systems that can be usedin the practice of the methods taught herein, according to someembodiments. The computer system 100 may be a conventional computersystem and includes a computer 105, I/O devices 110, and a displaydevice 115. The computer 105 can include a processor 120, acommunications interface 125, memory 130, display controller 135,non-volatile storage 140, and I/O controller 145. The computer system100 may be coupled to or include the I/O devices 150 and display device155.

The computer 105 interfaces to external systems through thecommunications interface 125, which may include a modem or networkinterface. It will be appreciated that the communications interface 125can be considered to be part of the computer system 100 or a part of thecomputer 105. The communications interface 125 can be an analog modem,isdn modem, cable modem, token ring interface, satellite transmissioninterface (e.g. “direct PC”), or other interfaces for coupling thecomputer system 100 to other computer systems. In a cellular telephoneor PDA, for example, this interface can be a radio interface forcommunication with a cellular network and may also include some form ofcabled interface for use with an immediately available personalcomputer. In a two-way pager, the communications interface 125 istypically a radio interface for communication with a data transmissionnetwork but may similarly include a cabled or cradled interface as well.In a personal digital assistant, for example, the communicationsinterface 125 typically can include a cradled or cabled interface andmay also include some form of radio interface, such as a BLUETOOTH or802.11 interface, or a cellular radio interface.

The processor 120 may be, for example, a conventional microprocessorsuch as an Intel Pentium microprocessor or Motorola power PCmicroprocessor, a Texas Instruments digital signal processor, or acombination of such components. The memory 130 is coupled to theprocessor 120 by a bus. The memory 130 can be dynamic random accessmemory (DRAM) and can also include static ram (SRAM). The bus couplesthe processor 120 to the memory 130, also to the non-volatile storage140, to the display controller 135, and to the I/O controller 145.

The I/O devices 150 can include a keyboard, disk drives, printers, ascanner, and other input and output devices, including a mouse or otherpointing device. The display controller 136 may control in theconventional manner a display on the display device 155, which can be,for example, a cathode ray tube (CRT) or liquid crystal display (LCD).The display controller 135 and the I/O controller 145 can be implementedwith conventional well known technology, meaning that they may beintegrated together, for example.

The non-volatile storage 140 is often a FLASH memory or read-onlymemory, or some combination of the two. Any non-volatile storage can beused. A magnetic hard disk, an optical disk, or another form of storagefor large amounts of data may also be used in some embodiments, althoughthe form factors for such devices typically preclude installation as apermanent component in some devices. Rather, a mass storage device onanother computer is typically used in conjunction with the more limitedstorage of some devices. Some of this data is often written, by a directmemory access process, into memory 130 during execution of software inthe computer 105. One of skill in the art will immediately recognizethat the terms “machine-readable medium,” “computer-readable storagemedium,” or “computer-readable medium” includes any type of storagedevice that is accessible by the processor 120 and also encompasses acarrier wave that encodes a data signal. Objects, methods, inlinecaches, cache states and other object-oriented components may be storedin the non-volatile storage 140, or written into memory 130 duringexecution of, for example, an object-oriented software program. In someembodiments, these media can include modules or engines, for example, inwhich the modules or engines are complete, in that they can include thesoftware, hardware, software/hardware combinations, and any othercomponents recognized by one of skill that enable their operability intheir functions as taught herein.

The computer system 100 is one example of many possible differentarchitectures. For example, personal computers based on an Intelmicroprocessor often have multiple buses, one of which can be an I/O busfor the peripherals and one that directly connects the processor 120 andthe memory 130 (often referred to as a memory bus). The buses areconnected together through bridge components that perform any necessarytranslation due to differing bus protocols.

In addition, the computer system 100 is controlled by operating systemsoftware which includes a file management system, such as a diskoperating system, which is part of the operating system software. Oneexample of an operating system software with its associated filemanagement system software is the family of operating systems known asWindows CEO and Windows® from Microsoft Corporation of Redmond, Wash.,and their associated file management systems. Another example ofoperating system software with its associated file management systemsoftware is the LINUX operating system and its associated filemanagement system. Another example of an operating system software withits associated file management system software is the PALM operatingsystem and its associated file management system. The file managementsystem is typically stored in the non-volatile storage 140 and causesthe processor 120 to execute the various acts required by the operatingsystem to input and output data and to store data in memory, includingstoring files on the non-volatile storage 140. Other operating systemsmay be provided by makers of devices, and those operating systemstypically will have device-specific features which are not part ofsimilar operating systems on similar devices. Similarly, WinCE® or PALMoperating systems may be adapted to specific devices for specific devicecapabilities. Other examples include Google's ANDROID, Apple's 10S,Nokia's SYMBIAN, RIM's BLACKBERRY OS, Samsung's BADA, Microsoft'sWINDOWS PHONE, Hewlett-Packard's WEBOS, and embedded Linux distributionssuch as MAEMO and MEEGO, and the like.

The computer system 100 may be integrated onto a single chip or set ofchips in some embodiments, and typically is fitted into a small formfactor for use as a personal device. Thus, it is not uncommon for aprocessor, bus, onboard memory, and display/I-O controllers to all beintegrated onto a single chip. Alternatively, functions may be splitinto several chips with point-to-point interconnection, causing the busto be logically apparent but not physically obvious from inspection ofeither the actual device or related schematics.

FIG. 2 illustrates a processor-memory diagram to describe components ofa system, according to some embodiments. The system 200 shown in FIG. 2can include, for example, a processor 205 and a memory 210 (that caninclude non-volatile memory), wherein the memory 210 includes asubject-profile module 215, a database 220, an offering module 225, asolutions module 230, an integration engine 235, and an instructionmodule 240. And, as shown in the figure, other components can beincluded.

The system includes an input device (not shown) operable to allow a userto enter a personalized subject-profile into the computing system.Examples of input devices include a keyboard, a mouse, a data exchangemodule operable to interact with external data formats,voice-recognition software, a hand-held device in communication with thesystem, and the like.

The offering module 225 can be embodied in a non-transitory computerreadable storage medium and operable for offering an opportunity formembers of a network community to provide a submission of input data,response data, or the like, to the network community. The instructionmodule 240 can be embodied in a non-transitory computer readable storagemedium and operable for providing instruction to a member of the networkcommunity regarding a criteria for making a submission of any type, orinteracting within the community in any way.

The player/challenge database 220 can be embodied in a non-transitorycomputer readable storage medium and operable to store a library ofusers, user-submissions, input data, response data, and the like,wherein the database can include any text or any other media, includingdata compilations, statistics, and the like, or whatever otherinformation may be considered useful to the network community.

The subject-profile module 215 can be embodied in a non-transitorycomputer readable storage medium and operable for receiving thepersonalized subject-profile and converting the personalized subjectprofile into a user profile. The user profile can comprise a set ofpersonal statistics for the user, along with a tracking of the user'sparticipation in the network community, as well as data regarding thesame. As such, this provides a way for users of similar interests toidentify one another and target community groups, subgroups, and evenone-on-one communications. The input device can allow a user to enter apersonalized subject-profile into a computing system. And, thepersonalized subject-profile can comprise a questionnaire designed toobtain information to be used to produce a personalized file for theuser.

The transformation module 230 can be embodied in a non-transitorycomputer readable storage medium and operable for parsing input data,response data, other such data, and the like in the database intocategories for use in user analyses. The enumeration engine 235 can beembodied in a non-transitory computer readable storage medium andoperable to parameterize, for example, a non-compartmental model forpredicting a test response to a test input.

It should be appreciated that any of the modules or engines can haveadditional functions, and additional modules or engines can be added tofurther provide even more functionality. Of course, the system will havea processor 205. And, the graphical user interface (not shown) can beused for displaying video, audio, and/or text to the user.

In some embodiments, the system further comprises a parameterizationmodule operable 245 to derive select parameters such as, for example,display-preference parameters from the user profile, and the graphicaluser interface displays select data from the database 220 in accordancewith the user's display preferences and in the form of the customizedset of information subset options. Select parameters may include userselections, administrator selections, or some combination thereof. Forexample, the user may prefer a select combination of shapes, colors,sound, and any other of a variety of screen displays and multimediaoptions. Furthermore, the selections can be used to personalize andchange the display-preference parameters easily and at any time.

In some embodiments, the system further comprises a data exchange module250 operable to interact with external data formats obtained fromanother database or source, such as a remote memory source, includingany external memory or file known to one of skill, including other userdatabases within the network community.

In some embodiments, the system further comprises a messaging module(not shown) operable to allow users to communicate with other users. Theusers can email one another, post blogs, or have instant messagingcapability for real-time communications. In some embodiments, the usershave video and audio capability in the communications, wherein thesystem implements data streaming methods known to those of skill in theart.

The systems taught herein can be practiced with a variety of systemconfigurations, including personal computers, multiprocessor systems,microprocessor-based or programmable consumer electronics, network PCs,minicomputers, mainframe computers, and the like. The teachings can alsobe practiced in distributed computing environments where tasks areperformed by remote processing devices that are linked through acommunications network. As such, in some embodiments, the system furthercomprises an external computer connection and a browser program module270. The browser program module 270 can be operable to access externaldata through the external computer connection.

FIG. 3 is a concept diagram illustrating a system taught herein,according to some embodiments. The system 300 contains components thatcan be used in a typical embodiment. In addition to the subject-profilemodule 215, database 220, the offering module 225, the transformationmodule 230, the enumeration engine 235, and the instruction module 240shown in FIG. 2, the memory 210 of the device 300 also includesparameterization module 245 and the browser program module 270 foraccessing the external database 320. The system can include a speaker352, display 353, and a printer 354 connected directly or through I/Odevice 350 connected to I/O backplane 340.

It should be appreciated that, in some embodiments, the system can beimplemented in a stand-alone device, rather than a computer system ornetwork, such that the device functions as a virtual system as providedherein, but does not perform any other substantially differentfunctions. In figure FIG. 3, for example, the I/O device 350 connects tothe speaker (spkr) 352, display 353, and microphone (mic) 354, but couldalso be coupled to other features. Other features can be added such as,for example, an on/off button, a start button, an ear phone input, andthe like. In some embodiments, the system can turn on and off throughmotion. And, in some embodiments, the systems can include securitymeasures to protect the user's privacy, integrity of data, or both.

State-of-the-Art Modeling is Complex, Insufficient, and Ambiguous

Input-response computer modeling is typically formulated mathematicallyby relating the rates of change of species within the system to amountsof species present in the system. Rates of change are expressed asfirst-order derivatives; therefore the resulting formulation is a systemof first-order differential equations. Running a simulation, or runningthe model, is simply solving the system of differential equations. Theoutput of the simulation are the concentration vs. time curves of eachof the species in the system. The coefficients of the terms in thedifferential equations are often referred to as the parameters of themodel. An example of such a system is given below:

$\frac{\partial C_{1}}{\partial t} = {{k_{11}C_{1}} + {k_{12}C_{2}} + \ldots\mspace{11mu} + {k_{1n}C_{n}}}$$\frac{\partial C_{2}}{\partial t} = {{k_{21}C_{1}} + {k_{22}C_{2}} + \ldots\mspace{11mu} + {k_{2n}C_{n}}}$⋮   ⋮${\frac{\partial C_{n}}{\partial t} = {{k_{n\; 1}C_{1}} + {k_{n\; 2}C_{2}} + \ldots\mspace{11mu} + {k_{nn}C_{n}}}};$

where, in this example, C₁, C₂, . . . , C_(n) represent theconcentrations of the n different species in the system and k₁₁, k₁₂, .. . , k_(nn) are the parameters of the model. Changing the values of theparameters will change the output of the model. Proper adjustment of theparameters will yield the desired output; i.e., concentration curvesthat match a desired set of available data. This adjustment ofparameters to produce desired output is referred to as parameteroptimization or model calibration.

If all of the k_(ij)'s are real-valued constants, then the system issaid to be a linear system of differential equations. Many physicalsystems are modeled using linear systems of differential equations, butthere are often cases where a linear model is insufficient and anonlinear model is required. In a nonlinear system of differentialequations, at least one of the k_(ij)'s is a function of one of theC_(i)'s. For a linear system, the solution for the C_(i)'s as functionsof time will be of the form:

$\begin{matrix}{{{C_{i}(t)} = {{M_{i_{1}}e^{\beta_{i_{1}}t}} + {M_{i_{2}}e^{\beta_{i_{2}}t}} + \ldots\mspace{14mu} + {M_{i_{n}}e^{\beta_{i_{n}}t}}}};} & (1)\end{matrix}$

where, the number of terms n is the same as the number of species beingmodeled. Each of the solution variables M_(ij) and β_(ij) is a functionof the model parameters k₁₁, . . . , k_(nn). For a linear system, eachof the solution functions, C_(i)(t), will be a linear function of allthe initial values, C_(i)(0).

This approach of setting up a model (system of differential equations,etc.) with associated parameters that affect the output (solutionfunctions) is called a mechanistic approach to modeling. In amechanistic approach, the model species and parameters can beconstructed to represent actual physiological components(physiologically-based modeling) or can simply serve as a sufficientnumber of mathematical degrees of freedom to allow for accurate modelfits to given data.

In order to formulate a system of differential equations in the modelingprocess, a compartmental approach is often used. That is, a network ofcompartments is set up, with connections between each that specify therate at which species are transferred between compartments.Compartmental models can be constructed using linear or nonlinearreactions between compartments. In linear models, parameter values areconstants. FIG. 1 shows an example of a simple two-compartment linearmodel, with forward (k_(f)) and reverse (k_(r)) reactions between thetwo compartments as well as elimination (k_(e)) from the secondcompartment. In this linear model, k_(f), k_(r), and k_(e) are allreal-valued constants.

FIG. 4 shows an example of a prior art, simple two-compartment linearmodel, with forward (k_(f)) and reverse (k_(r)) reactions between thetwo compartments as well as elimination (k_(e)) from the secondcompartment, according to some embodiments.

The resulting differential equations are:

${V_{1}\frac{\partial C_{1}}{\partial t}} = {{{- k_{f}}C_{1}} + {k_{r}C_{2}}}$${{V_{2}\frac{\partial C_{2}}{\partial t}} = {{k_{f}C_{1}} - {( {k_{r} + k_{e}} )C_{2}}}};$

where, V₁ and V₂ represent the physical volumes of compartments 1 and 2,respectively. These volumes are often not known and have to be eitherphysically or mathematically estimated. The compartmental modelingapproach can be, but is not always, physiologically-based. In aphysiologically-based model, each compartment represents an actualphysiological entity, and the reactions between compartments are basedon expert knowledge of the interactions between the includedphysiological entities.

FIG. 4 is an example of a mechanistic approach to input-responsemodeling, and the vast majority of input-response modeling is done usinga mechanistic approach. In this approach, the components of themodel—nodes, connections, differential equations, parameters, etc.—areset up based on knowledge of the underlying physical mechanisms presentin the system. Parameter values are initially set based on expertknowledge of how certain components of the system should behave withrespect to other related components. This provides a very useful tool inexploratory research, where one can examine the effects that result fromturning certain ‘knobs’ or ‘handles’ (parameters) in the model. Thereare two serious limitations of this mechanistic approach. The first isone of sufficiency and the other is one of ambiguity.

Mechanistic models often lack sufficient content to provide accuratepredictions of input-response relationships, and this is because currentexpert knowledge is often lacking in its ability to fully characterize asystem or all of the interactions within a system. This lack ofknowledge might manifest itself in not having enough compartments in acompartmental model, or in having linear transfer rates betweencompartments when in fact the underlying process is nonlinear. What isoften done in these cases is to go back to the model and arbitrarily addcompartments or make certain reactions nonlinear, in an attempt providethe necessary mathematical foundation to allow for sufficiently accuratefits to given data. In this way, many models becomenon-physiologically-based when the intent was to build aphysiologically-based model.

Another significant limitation to the mechanistic approach comes fromthe fact that in mechanistic models, the model parameters are serving asan intermediary between the model inputs and outputs. The parameters areuseful in serving as handles to affect output, but there is often not aunique mapping between model inputs and output. That is, there may bemore than one way (or even an infinite number of ways) to achieve acertain output from a given set of inputs. This ambiguity can be veryproblematic when attempting to do things like map the properties of theinput to the output. For example, in a dose-response model, it would beextremely valuable to be able to map molecular properties of a drugcompound to a particular response within the body. Using a mechanisticdose-response model, this mapping would have to go from input to modelparameters to output. If there are many different sets of modelparameters that can produce the same output, then it becomes verydifficult, or impossible, to use the parameters as an intermediary inconstructing an effective mapping from input properties (molecularproperties of dose compound) to output (response within the body).

The Systems and Methods Set-Forth Herein are Simple, Sufficient, andUnambiguous

To address the limitations of the current, state-of-the-art, theteachings set-forth herein include a novel system of modeling that usesa data-based, non-mechanistic, differential-equation-free approach forpredicting a particular response of a system to a given input. There isno system of differential equations, yet the form of the responsefunction is similar to a solution function obtained from a system ofdifferential equations. Because there is no system of differentialequations, there are no associated “model parameters.” The only unknownsthat need to be optimized are the variables in the response function.This eliminates the potential ambiguity that is present in usingdifferential equation parameters as the intermediary between input andoutput, as is the case in a mechanistic approach. The response functionfor this new approach is an extension of the solution function for asystem of linear differential equations, Equation 1, where theexponential terms are replaced by terms containing rational functions ofexponentials. The basic form is given by:

$\begin{matrix}{{C(t)} = {M_{0} + {M_{1}\lbrack \frac{1 - e^{{- \alpha_{1}}t}}{1 + {( {e^{K} - 2} )e^{{- \alpha_{1}}t}}} \rbrack} + \ldots + {{M_{n}\lbrack \frac{1 - e^{{- \alpha_{n}}t}}{1 + {( {e^{K} - 2} )e^{{- \alpha_{n}}t}}} \rbrack}.}}} & (2)\end{matrix}$

If K=ln(2), then the response function (2) reduces to a form that isequivalent to the linear solution function (1).

One of the characteristics of a solution function for a nonlinear systemis that the variables M₀, M₁, . . . , M_(n) and α₁, . . . , α_(n) areall functions of the initial input condition, or dose. That is, if wedefine dose as C₀, then M₀, M₁, . . . , M_(n) and α₁, . . . , α_(n) areall functions of C₀. In this new formulation, the functions M₀(C₀),M₁(C₀), . . . , M_(n)(C₀) and α₁(C₀), . . . , α_(n)(C₀) are also definedusing the formulation of Equation (2). These functions are given by:

${M_{0}( C_{0} )} = {M_{0}^{0} + {M_{0}^{1}\lbrack \frac{1 - e^{{({- \alpha_{M_{0}}^{1}})}C_{0}}}{1 + {( {e^{K_{M_{0}}} - 2} )e^{{({- \alpha_{M_{0}}^{1}})}C_{0}}}} \rbrack} + \ldots + {M_{0}^{q}\lbrack \frac{1 - e^{{({- \alpha_{M_{0}}^{q}})}C_{0}}}{1 + {( {e^{K_{M_{0}}} - 2} )e^{({- \alpha_{M_{0}}^{q}})}C_{0}}} \rbrack}}$⋮${M_{n}( C_{0} )} = {{M_{n}^{0} + {M_{n}^{1}\lbrack \frac{1 - e^{{({- \alpha_{M_{n}}^{1}})}C_{0}}}{1 + {( {e^{K_{M_{n}}} - 2} )e^{({- \alpha_{M_{n}}^{1}})}C_{0}}} \rbrack} + \ldots + {{M_{0}^{q}\lbrack \frac{1 - e^{{({- \alpha_{M_{n}}^{q}})}C_{0}}}{1 + {( {e^{K_{M_{n}}} - 2} )e^{({- \alpha_{M_{n}}^{q}})}C_{0}}} \rbrack}{\alpha_{1}( C_{0} )}}} = {{N_{1}^{0} + {N_{1}^{1}\lbrack \frac{1 - e^{{({- \alpha_{\alpha_{1}}^{1}})}C_{0}}}{1 + {( {e^{K_{\alpha_{1}}} - 2} )e^{({- \alpha_{\alpha_{1}}^{1}})}C_{0}}} \rbrack} + \ldots + {{N_{1}^{q}\lbrack \frac{1 - e^{{({- \alpha_{\alpha_{1}}^{q}})}C_{0}}}{1 + {( {e^{K_{\alpha_{1}}} - 2} )e^{({- \alpha_{\alpha_{1}}^{q}})}C_{0}}} \rbrack}\vdots{\alpha_{n}( C_{0} )}}} = {N_{n}^{0} + {N_{n}^{1}\lbrack \frac{1 - e^{{({- \alpha_{\alpha_{n}}^{1}})}C_{0}}}{1 + {( {e^{K_{\alpha_{n}}} - 2} )e^{({- \alpha_{\alpha_{n}}^{1}})}C_{0}}} \rbrack} + \ldots + {N_{n}^{q}\lbrack \frac{1 - e^{{({- \alpha_{\alpha_{n}}^{q}})}C_{0}}}{1 + {( {e^{K_{\alpha_{n}}} - 2} )e^{({- \alpha_{\alpha_{n}}^{q}})}C_{0}}} \rbrack}}}}$

The full implementation of this formulation would require the estimationof a large number of parameters. In many embodiments, however, a reducedform will be sufficient for providing accurate models of input-responserelationships. In some embodiments, a less reduced form, or even thefull implementation, may be used.

The reduced form makes two assumptions. The first is that the number ofterms in the M₀(C₀), M₁(C₀), . . . , M_(n)(C₀) and α₁(C₀), . . . ,α_(n)(C₀) functions is truncated at 1; i.e., q=1. The second is thatonly one α parameter and only one K parameter is used for all of theM₀(C₀), M₁(C₀), . . . , M_(n)(C₀) and α₁(C₀), . . . , α_(n)(C₀)functions; i.e.,

$\begin{matrix}{\alpha_{M_{0}}^{1} = {\alpha_{M_{1}}^{1} = {\ldots = {\alpha_{M_{n}}^{1} = {\alpha_{\alpha_{1}}^{1} = {\ldots = {\alpha_{\alpha_{n}}^{1} \equiv \alpha_{p}}}}}}}} & (3) \\{K_{M_{0}}^{1} = {K_{M_{1}}^{1} = {\ldots = {K_{M_{n}}^{1} = {K_{\alpha_{1}}^{1} = {\ldots = {K_{\alpha_{n}}^{1} \equiv K_{p}}}}}}}} & (4)\end{matrix}$

Substituting the relationships (3) and (4) into the functions M₀(C₀),M₁(C₀), . . . , M_(n)(C₀) and α₁(C₀), . . . , α_(n)(C₀), and truncatingthose functions at q=1 yields:

$\begin{matrix}{{{M_{0}( C_{0} )} = {M_{0}^{0} + {M_{0}^{1}\lbrack \frac{1 - e^{{- \alpha_{p}}C_{0}}}{1 + {( {e^{K_{p}} - 2} )e^{{- \alpha_{p}}C_{0}}}} \rbrack}}}\vdots} & (5) \\{{M_{n}( C_{0} )} = {M_{n}^{0} + {M_{n}^{1}\lbrack \frac{1 - e^{{- \alpha_{p}}C_{0}}}{1 + {( {e^{K_{p}} - 2} )e^{{- \alpha_{p}}C_{0}}}} \rbrack}}} & (6) \\{{{\alpha_{1}( C_{0} )} = {N_{1}^{0} + {N_{1}^{1}\lbrack \frac{1 - e^{{- \alpha_{p}}C_{0}}}{1 + {( {e^{K_{p}} - 2} )e^{{- \alpha_{p}}C_{0}}}} \rbrack}}}\vdots} & (7) \\{{\alpha_{n}( C_{0} )} = {N_{n}^{0} + {N_{n}^{1}\lbrack \frac{1 - e^{{- \alpha_{p}}C_{0}}}{1 + {( {e^{K_{p}} - 2} )e^{{- \alpha_{p}}C_{0}}}} \rbrack}}} & (8)\end{matrix}$

To simplify, define a kernel function, which is a function of initialinput condition, or dose, C₀

${kernel} \equiv {\frac{1 - e^{{- \alpha_{p}}C_{0}}}{1 + {( {e^{K_{p}} - 2} )e^{{- \alpha_{p}}C_{0}}}}.}$

Substituting Equations (5)-(8) into Equation (2) yields:

$\begin{matrix}{{C(t)} = {\lbrack {M_{0}^{0} + {M_{0}^{1}({kernel})}} \rbrack + {\quad{{\lbrack {M_{1}^{0} + {M_{1}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}{1 - {( {e^{K} - 2} )e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}} \}} + \ldots + {\quad{\lbrack {M_{n}^{0} + {M_{n}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}{1 - {( {e^{K} - 2} )e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}} \}}}}}}} & (9)\end{matrix}$

This new form for the response function allows for nonlinear modelbehavior as well as time-lagged effects. Theoretically, this allows foraccurate characterization and model description of complex physicalphenomena. The new response function contains n terms, where n is anarbitrary number and can be set to achieve desired accuracy.

The modeling approach using this new formulation is to estimate thevalues of K, K_(p), α_(p), M₀ ⁰, . . . , M_(n) ⁰, M₀ ¹, . . . , M_(n) ¹,N₁ ⁰, . . . , N_(n) ⁰, and N₁ ¹, . . . , N_(n) ¹ that yield the best fitof response function to available data. In this approach, there will bemuch less (ideally not any) ambiguity between values of responsefunction variables and goodness of fit between model and data. In otherwords, if you define error as the difference between available data andmodel prediction, then error as a function of response functionvariables will be more convex and contain fewer local minima than theerror as a function of model parameters in the case of a mechanisticmodeling approach.

A great deal of system information is condensed into the responsefunction variables of the new formulation. Complex phenomena such asnonlinear behavior can be described using much fewer degrees of freedomthan is the case in a mechanistic approach where a large number of modelparameters is typically used. This will reduce the redundancy that oftenoccurs in mechanistic models using a large number of model parameters.The response variables in the new formulation can even take into accountinformation that is not known prior to building a model, but shows up inthe form of response data. Thus, the new formulation avoids theinsufficiency that is often seen in mechanistic models.

Essentially, the variables in the response function (Equation (9)) willall be unique functions of the model parameters, but the reverse is nottrue. That is, the model parameters are not necessarily unique functionsof response variables (as will be demonstrated in detail in Example 1).Therefore, the response variables in the systems and methods taughtherein represent some (unknown) function of model parameters, if therewere model parameters. But because the systems and methods taught hereinallow for nonlinear behavior, for which there are not analyticalsolutions, the response variables represent complicated functions ofmany different potential model parameters, and therefore providesufficiency in the case where sufficient knowledge does not exist to apriori build the model and its parameters. This new formulation alsoremoves the ambiguity that exists in mechanistic modeling approaches,where model parameters are not unique functions of response variables.

The optimization of the response variables in the systems and methodstaught herein is even more complicated than the optimization of thevariables in the linear response function given by Equation (1), andrequires a series of unconstrained and constrained linear and nonlinearoptimization procedures (which are described in more detail in Example9). It should be noted that if a linear response function is sufficient,then the optimization of the response variables in the systems andmethods taught herein can be used, in some embodiments, to yield aresponse function that is equivalent to the linear response functiongiven by Equation (1). Therefore, the systems and methods taught hereinwill accurately describe both linear and nonlinear phenomena. Onceoptimal values of response variables are obtained for a given system,the model can be used to yield an accurate prediction of the system'sresponse to the introduction of an input of interest. The goal of thismethod is to provide accurate input-response predictions over a widerange of scale. For example, this algorithm could be used to makeaccurate predictions of responses on the tissue/organ-scale in the humanbody based solely on the molecular properties of input compounds. Thiscould have significant impact in areas such asabsorption-distribution-metabolism-excretion (ADME) prediction in drugdesign, as well as drug development in personalized medicine.

FIG. 5 illustrates a flowchart for a non-compartmental method ofpredicting a time-dependent response of a component of a system to aninput into the system, according to some embodiments. The method cancomprise identifying 505 the system and the component, identifying 510the input, and identifying 515 the time-dependent response; wherein, theinput includes a set of actual 520 inputs and a test 525 input, and thetime-dependent response includes a set of time-dependent actual 530responses and a test 535 response; obtaining the set of time-dependentactual responses of the component to the set of actual inputs; and,using the set of actual inputs and the set of time-dependent actualresponses to provide 540 a model for predicting the test response to thetest input, the model comprising the formula:

$\begin{matrix}{{C(t)} = {\lbrack {M_{0}^{0} + {M_{0}^{1}({kernel})}} \rbrack + {\quad{{\lbrack {M_{1}^{0} + {M_{1}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}{1 - {( {e^{K} - 2} )e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}} \}} + \ldots + {\quad{\lbrack {M_{n}^{0} + {M_{n}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}{1 - {( {e^{K} - 2} )e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}} \}}}}}}} & (9)\end{matrix}$

-   -   wherein,    -   M⁰ ₀, . . . , M⁰ _(n) and M¹ ₀, . . . , M¹ _(n) are overall        scaling parameters;    -   N⁰ ₁, . . . , N⁰ _(n) and N¹ ₁, . . . , N¹ _(n) are exponential        scaling parameters;    -   n ranges from 1 to 4;    -   K is an overall shifting parameter; and,    -   C(t) is the time-dependent response to the test input at time t;    -   and,

${{kernel} \equiv \frac{1 - e^{{- \alpha_{p}}C_{0}}}{1 + {( {e^{K_{p}} - 2} )e^{{- \alpha_{p}}C_{0}}}}};$

-   -   wherein, C₀ is the initial amount of the test input; K_(p) is a        shifting parameter related to C₀; and, α_(p) is shifting and        scaling parameter related to C₀.

The last step in FIG. 5 is using 550 the model for predictions.

Non-compartmental methods of predicting a time-dependent response of acomponent of a mammalian system to an input into the system are alsoprovided. In these embodiments, the methods can comprise selecting acomponent of the system, the component selected from the groupconsisting of a cell, a tissue, an organ, a DNA, a virus, a protein, anantibody, a bacteria; selecting a set of actual inputs, the set ofactual inputs having an element selected from the group consisting of aDNA, a virus, a protein, an antibody, a bacteria, a chemical, a dietarysupplement, a nutrient, and a drug; obtaining a set of time-dependentactual responses of the component to the set of actual inputs; and,using the set of actual inputs and the set of time-dependent actualresponses to provide a model for predicting a test response to a testinput, the model comprising the formula

$\begin{matrix}{{C(t)} = {\lbrack {M_{0}^{0} + {M_{0}^{1}({kernel})}} \rbrack + {\quad{{\lbrack {M_{1}^{0} + {M_{1}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}{1 - {( {e^{K} - 2} )e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}} \}} + \ldots + {\quad{\lbrack {M_{n}^{0} + {M_{n}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}{1 - {( {e^{K} - 2} )e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}} \}}}}}}} & (9)\end{matrix}$

-   -   wherein,    -   M⁰ ₀, . . . , M⁰ _(n) and M¹ ₀, . . . , M¹ _(n) are overall        scaling parameters;    -   N⁰ ₁, . . . , N⁰ _(n) and N¹ ₁, . . . , N¹ _(n) are exponential        scaling parameters;    -   n ranges from 1 to 4;    -   K is an overall shifting parameter; and,    -   C(t) is the time-dependent response to the test input at time t;    -   and,

${{kernel} \equiv \frac{1 - e^{{- \alpha_{p}}C_{0}}}{1 + {( {e^{K_{p}} - 2} )e^{{- \alpha_{p}}C_{0}}}}};$

-   -   wherein, C₀ is the initial amount of the test input; K_(p) is a        shifting parameter related to C₀; and, α_(p) is shifting and        scaling parameter related to C₀.

Devices for predicting a time-dependent response of a component of aphysical system to an input into the system are provided. In theseembodiments, the device can comprise a processor; a database for storinga set of actual input data, a set of time-dependent actual responsedata, test input data, and time-dependent test response data on anon-transitory computer readable medium; an enumeration engine on anon-transitory computer readable medium to parameterize anon-compartmental model for predicting a test response to a test input,the non-compartmental model comprising the formula

$\begin{matrix}{{C(t)} = {\lbrack {M_{0}^{0} + {M_{0}^{1}({kernel})}} \rbrack + {\quad{{\lbrack {M_{1}^{0} + {M_{1}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}{1 - {( {e^{K} - 2} )e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}} \}} + \ldots + {\quad{\lbrack {M_{n}^{0} + {M_{n}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}{1 - {( {e^{K} - 2} )e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}} \}}}}}}} & (9)\end{matrix}$

-   -   wherein,    -   M⁰ ₀, . . . , M⁰ _(n) and M¹ ₀, . . . , M¹ _(n) are overall        scaling parameters;    -   N⁰ ₁, . . . , N⁰ _(n) and N¹ ₁, . . . , N¹ _(n) are exponential        scaling parameters;    -   n ranges from 1 to 4;    -   K is an overall shifting parameter; and,    -   C(t) is the time-dependent response to the test input at time t;    -   and,

${{kernel} \equiv \frac{1 - e^{{- \alpha_{p}}C_{0}}}{1 + {( {e^{K_{p}} - 2} )e^{{- \alpha_{p}}C_{0}}}}};$

-   -   wherein, C₀ is the initial amount of the test input; K_(p) is a        shifting parameter related to C₀; and, α_(p) is shifting and        scaling parameter related to C₀;        and, a transformation module on a non-transitory computer        readable medium to transform the test data into the        time-dependent response data using the non-compartmental model.

The systems can be virtually any physical or non-physical system knownto one of skill in which that person of skill may want to predict aparticular response of the system to a given input. In some embodiments,the system can be an environmental system, and the component can beselected from the group consisting of air, water, and soil. In someembodiments, the system can be a mammal, and the component can beselected from the group consisting of a cell, a tissue, an organ, a DNA,a virus, a protein, an antibody, a bacteria. In some embodiments, thesystem can be a chemical system, a biological system, a mechanicalsystem, an electrical system, a financial system, a sociological system,a political system, or a combination thereof. As such, the teachingsprovided herein include general methods of predicting a particularresponse of any such system to a given input. For example, a biologicalsystem can have a biological input, a mechanical system can have amechanical data input, an electrical system can have a relativeelectrical data input, a financial system can have a relative financialdata input, a sociological system can have a relative sociological datainput, a political system can have a relative political data input, andthe like.

In some embodiments, the input into the system can cause a substantialeffect or a negligible effect. The term “negligible effect” can be used,for example, to mean that the activity does not increase or decreasemore than about 10% when compared to any one or any combination of thecompounds of interest, respectively, without the other components. Insome embodiments, the term “negligible effect” can be used to refer to achange of less that 10%, less than 9%, less than 8%, less than 7%, lessthan 6%, less than 5%, less than 4%, and less than 3%. In someembodiments, the term “negligible effect” can be used to refer to achange ranging from about 3% to about 10%, in increments of 1%.

The effects of the input can be biological, such as in drug testing orthe testing of compositions used in treating a subject. The compositionstested, for example, can be referred to as extracts, compositions,compounds, agents, active agents, bioactive agents, supplements, drugs,and the like. In some embodiments, the terms “composition,” “compound,”“agent,” “active”, “active agent”, “bioactive agent,” “supplement,” and“drug” can be used interchangeably and, it should be appreciated that, a“formulation” can comprise any one or any combination of these.Likewise, in some embodiments, the composition can also be in a liquidor dry form, where a dry form can be a powder form in some embodiments,and a liquid form can include an aqueous or non-aqueous component.Moreover, the term “bioactivity” can refer to the function of thecompound when administered in any way known to one of skill, includingparenterally or non-parenterally, including orally, topically, orrectally to a subject. In some embodiments, the term “target site” canbe used to refer to a select location on or in a subject that couldbenefit from an administration of a compound. In some embodiments, atarget can include any site of action in which the agent's activity,such as any therapeutic activity including anti-hyproliferativeactivity, antioxidant activity, anti-inflammatory activity, analgesicactivity, and the like, can serve a benefit to the subject. The targetsite can be a healthy or damaged tissue of a subject. As such, theteachings include a method of administering one or more compounds taughtherein to any healthy or damaged tissue, such as epithelial, connective,muscle, or nervous tissue, including hematopoietic, dermal, mucosal,gastrointestinal or otherwise.

The systems and methods herein can determine the stability of acomposition in a system. In some embodiments, a composition orformulation can be considered as “stable” if it loses less than 10% ofits original activity. In some embodiments, a composition or formulationcan be considered as stable if it loses less than 5%, 3%, 2%, or 1% ofits original activity. In some embodiments, a composition or formulationcan be considered as “substantially stable” if it loses greater thanabout 10% of its original activity, as long as the composition canperform it's intended use to a reasonable degree of efficacy. In someembodiments, the composition can be considered as substantially stableif it loses activity at an amount greater than about 12%, about 15%,about 25%, about 35%, about 45%, about 50%, about 60%, or even about70%. The activity loss can be measured by comparing activity at the timeof packaging to the activity at the time of administration, and this caninclude a reasonable shelf life. In some embodiments, the composition isstable or substantially stable, if it remains useful for a periodranging from 3 months to 3 years, 6 months to 2 years, 1 year, or anytime period therein in increments of about 1 month.

Moreover, the systems and methods provided herein can be used inpredicting the efficacy of therapeutic treatments. The terms “treat,”“treating,” and “treatment” can be used interchangeably in someembodiments and refer to the administering or application of thecompositions and formulations taught herein, including suchadministration as a health or nutritional supplement, and alladministrations directed to the prevention, inhibition, amelioration ofthe symptoms, or even a cure of a condition in a subject. The terms“disease,” “condition,” “disorder,” and “ailment” can be usedinterchangeably in some embodiments. The term “subject” and “patient”can be used interchangeably in some embodiments and refer to an animalsuch as a mammal including, but not limited to, non-primates such as,for example, a cow, pig, horse, cat, dog, rat and mouse; and primatessuch as, for example, a monkey or a human. As such, the terms “subject”and “patient” can also be applied to non-human biologic applicationsincluding, but not limited to, veterinary, companion animals, commerciallivestock, and the like.

In some embodiments, the methods further comprise orally administeringan effective amount of an oral dosage form of a composition to a subjectto systemically treat a disease or disorder, including any disease ordisorder taught herein. In some embodiments, the methods furthercomprise orally administering an effective amount of an oral dosage formof a composition to a subject as a dietary supplement. In someembodiments, the methods further comprise orally administering aneffective amount of an oral dosage form of a composition to a subject incombination with a second drug. In some embodiments, the teachings aredirected to a method of treating an inflammation of a tissue of subject,the method comprising administering an effective amount of a compositionto a tissue of the subject. In some embodiments, the teachings aredirected to treating a wounded tissue, the method comprisingadministering an effective amount of a composition to a tissue of thesubject. In some embodiments, the teachings are directed to treating ahyperproliferative disorder, such as cancer, either liquid or solid, themethod comprising administering an effective amount of a composition toa subject in need thereof.

An “effective amount” of a compound can be used to describe atherapeutically effective amount or a prophylactically effective amount.An effective amount can also be an amount that ameliorates the symptomsof a disease. A “therapeutically effective amount” can refer to anamount that is effective at the dosages and periods of time necessary toachieve a desired therapeutic result and may also refer to an amount ofactive compound, prodrug or pharmaceutical agent that elicits anybiological or medicinal response in a tissue, system, or subject that issought by a researcher, veterinarian, medical doctor or other clinicianthat may be part of a treatment plan leading to a desired effect. Insome embodiments, the therapeutically effective amount should beadministered in an amount sufficient to result in amelioration of one ormore symptoms of a disorder, prevention of the advancement of adisorder, or regression of a disorder. In some embodiments, for example,a therapeutically effective amount can refer to the amount of an agentthat provides a measurable response of at least 5%, at least 10%, atleast 15%, at least 20%, at least 25%, at least 30%, at least 35%, atleast 40%, at least 45%, at least 50%, at least 55%, at least 60%, atleast 65%, at least 70%, at least 75%, at least 80%, at least 85%, atleast 90%, at least 95%, or at least 100% of a desired action of thecomposition.

In cases of the prevention or inhibition of the onset of a disease ordisorder, or where an administration is considered prophylactic, aprophylactically effective amount of a composition or formulation taughtherein can be used. A “prophylactically effective amount” can refer toan amount that is effective at the dosages and periods of time necessaryto achieve a desired prophylactic result, such as prevent the onset of asunburn, an inflammation, allergy, nausea, diarrhea, infection, and thelike. Typically, a prophylactic dose is used in a subject prior to theonset of a disease, or at an early stage of the onset of a disease, toprevent or inhibit onset of the disease or symptoms of the disease. Aprophylactically effective amount may be less than, greater than, orequal to a therapeutically effective amount.

In some embodiments, a therapeutically or prophylactically effectiveamount of a composition may range in concentration from about 0.01 nM toabout 0.10 M; from about 0.01 nM to about 0.5 M; from about 0.1 nM toabout 150 nM; from about 0.1 nM to about 500 μM; from about 0.1 nM toabout 1000 nM, 0.001 μM to about 0.10 M; from about 0.001 μM to about0.5 M; from about 0.01 μM to about 150 μM; from about 0.01 μM to about500 μM; from about 0.01 μM to about 1000 nM, or any range therein. Insome embodiments, the compositions may be administered in an amountranging from about 0.005 mg/kg to about 100 mg/kg; from about 0.005mg/kg to about 400 mg/kg; from about 0.01 mg/kg to about 300 mg/kg; fromabout 0.01 mg/kg to about 250 mg/kg; from about 0.1 mg/kg to about 200mg/kg; from about 0.2 mg/kg to about 150 mg/kg; from about 0.4 mg/kg toabout 120 mg/kg; from about 0.15 mg/kg to about 100 mg/kg, from about0.15 mg/kg to about 50 mg/kg, from about 0.5 mg/kg to about 10 mg/kg, orany range therein, wherein a human subject is often assumed to averageabout 70 kg. Moreover, the systems and methods taught herein can usemicro-dosing, which can include the administration of dosages that areone, two, or perhaps three orders of magnitude less than the dosagesdescribed above, in some embodiments.

Any drug activity can be investigated using the systems and methodstaught herein. In some embodiments, the activity can include, forexample, free radical scavenger and antioxidant, inhibiting lipidperoxidation and oxidative DNA damage; anti-inflammatory activity;neurological treatments for Alzheimer's disease (anti-amyloid and othereffects), Parkinson's disease, and other neurological disorders;anti-arthritic treatment; anti-ischemic treatment; treatments formultiple myeloma and myelodysplastic syndromes; psoriasis treatments(topically and orally); cystic fibrosis treatments; treatments for liverinjury and alcohol-induced liver disease; multiple sclerosis treatments;antiviral treatments, including human immunodeficiency virus (HIV)therapy; treatments of diabetes; cancer treatments; and, reducing riskof heart disease; to name a few.

Any response can be investigated using the systems and methods taughtherein. For example, the amounts of the agents can be reduced, evensubstantially, such that the amount of the agent or agents desired isreduced to the extent that a significant response is observed from thesubject. A “significant response” can include, but is not limited to, areduction or elimination of a symptom, a visible increase in a desirabletherapeutic effect, a faster response to the treatment, a more selectiveresponse to the treatment, or a combination thereof. In someembodiments, the other therapeutic agent can be administered, forexample, in an amount ranging from about 0.1 μg/kg to about 1 mg/kg,from about 0.5 μg/kg to about 500 μg/kg, from about 1 μg/kg to about 250μg/kg, from about 1 μg/kg to about 100 μg/kg from about 1 μg/kg to about50 μg/kg, or any range therein. Combination therapies can beadministered, for example, for 30 minutes, 1 hour, 2 hours, 4 hours, 8hours, 12 hours, 18 hours, 1 day, 2 days, 3 days, 4 days, 5 days, 6days, 7 days, 8 days, 9 days, 10 days, 2 weeks, 3 weeks, 4 weeks, 6weeks, 3 months, 6 months 1 year, any combination thereof, or any amountof time considered desirable by one of skill. The agents can beadministered concomitantly, sequentially, or cyclically to a subject.Cycling therapy involves the administering a first agent for apredetermined period of time, administering a second agent or therapyfor a second predetermined period of time, and repeating this cyclingfor any desired purpose such as, for example, to enhance the efficacy ofthe treatment. The agents can also be administered concurrently. Theterm “concurrently” is not limited to the administration of agents atexactly the same time, but rather means that the agents can beadministered in a sequence and time interval such that the agents canwork together to provide additional benefit. Each agent can beadministered separately or together in any appropriate form using anyappropriate means of administering the agent or agents. One of skill canreadily select the frequency, duration, and perhaps cycling of eachconcurrent administration.

As such, in some embodiments, the teachings are directed to a device forpredicting a time-dependent response of a component of a mammaliansystem to an input into the system. In these embodiments, the device cancomprise a processor; a database for storing a set of actual input data,a set of time-dependent actual response data, test input data, andtime-dependent test response data on a non-transitory computer readablemedium; an enumeration engine on a non-transitory computer readablemedium to parameterize a non-compartmental model for predicting a testresponse to a test input, the non-compartmental model comprising theformula

$\begin{matrix}{{C(t)} = {\lbrack {M_{0}^{0} + {M_{0}^{1}({kernel})}} \rbrack + {\quad{{\lbrack {M_{1}^{0} + {M_{1}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}{1 - {( {e^{K} - 2} )e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}} \}} + \ldots + {\quad{\lbrack {M_{n}^{0} + {M_{n}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}{1 - {( {e^{K} - 2} )e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}} \}}}}}}} & (9)\end{matrix}$

-   -   wherein,    -   M⁰ ₀, . . . , M⁰ _(n) and M¹ ₀, . . . , M¹ _(n) are overall        scaling parameters;    -   N⁰ ₁, . . . , N⁰ _(n) and N¹ ₁, . . . , N¹ _(n) are exponential        scaling parameters;    -   n ranges from 1 to 4;    -   K is an overall shifting parameter; and,    -   C(t) is the time-dependent response to the test input at time t;    -   and,

${{kernel} \equiv \frac{1 - e^{{- \alpha_{p}}C_{0}}}{1 + {( {e^{K_{p}} - 2} )e^{{- \alpha_{p}}C_{0}}}}};$

-   -   wherein, C₀ is the initial amount of the test input; K_(p) is a        shifting parameter related to C₀; and, α_(p) is shifting and        scaling parameter related to C₀;        and, a transformation module on a non-transitory computer        readable medium to transform the test data into the        time-dependent response data using the non-compartmental model.

Any desired component known to one of skill can be used, in which thedesired component is a component of interest to the person of skill. Insome embodiments, the component can be blood, a tumor cell, a virus, abacteria, or a combination thereof.

Any desired test response known to one of skill can be used, in whichthe desired test response is a response of interest to the person ofskill. In some embodiments, the test response is a bacterial load, aviral load, a tumor marker, a blood chemistry, or a combination thereof.

Any desired set of actual inputs known to one of skill can be used, inwhich the desired set of actual inputs are of interest to the person ofskill. In some embodiments, the set of actual inputs can include a setof dosages of a drug, a set of drugs, or a combination thereof.

Any desired input known to one of skill can be used, in which thedesired input is of interest to the person of skill. For example, thesystems, methods, and devices can be used in drug screening. In someembodiments, the input is a diabetes drug candidate, and thetime-dependent response can be glucose in the bloodstream. In someembodiments, the input is a cancer drug candidate, and thetime-dependent response can be a cell apoptosis, tumor size reduction,reduced metastasis. In some embodiments, the input is an antibiotic drugcandidate, and the time-dependent response can be a bacterial load. Insome embodiments, the input is an antiviral drug candidate, and thetime-dependent response can be a viral load. In some embodiments, theinput is an immunomodulatory drug candidate, and the time-dependentresponse can be a measure of an immune response. In some embodiments,the input is an anti-inflammatory drug candidate, and the time-dependentresponse can be an inflammatory response. In some embodiments, the inputis an analgesic drug candidate, and the time-dependent response can be apain response.

The systems, methods, and devices taught herein transform input datainto response data and, as such, can be used to obtain thetime-dependent test response to the test input. And, the devices taughtherein can be in any form, whether handheld, desktop, intranet,internet, or otherwise cloud-based. In some embodiments, the device canbe a handheld device including, but not limited to, a PDA, a smartphone,an iPAD, a personal computer, and the like, including devices that arenot intended for any other substantial use.

FIG. 6 shows how a network may be used for the systems and methodstaught herein, in some embodiments. FIG. 6 shows several computersystems coupled together through a network 605, such as the internet,along with a cellular network and related cellular devices. The term“internet” as used herein refers to a network of networks which usescertain protocols, such as the TCP/IP protocol, and possibly otherprotocols such as the hypertext transfer protocol (HTTP) for hypertextmarkup language (HTML) documents that make up the world wide web (web).The physical connections of the internet and the protocols andcommunication procedures of the internet are well known to those ofskill in the art.

Access to the internet 605 is typically provided by internet serviceproviders (ISP), such as the ISPs 610 and 615. Users on client systems,such as client computer systems 630, 650, and 660 obtain access to theinternet through the internet service providers, such as ISPs 610 and615. Access to the internet allows users of the client computer systemsto exchange information, receive and send e-mails, and view documents,such as documents which have been prepared in the HTML format, forexample. These documents are often provided by web servers, such as webserver 620 which is considered to be “on” the internet. Often these webservers are provided by the ISPs, such as ISP 610, although a computersystem can be set up and connected to the internet without that systemalso being an ISP.

In some embodiments, the system is a web enabled application and canuse, for example, Hypertext Transfer Protocol (HTTP) and HypertextTransfer Protocol over Secure Socket Layer (HTTPS). These protocolsprovide a rich experience for the end user by utilizing web 2.0technologies, such as AJAX, Macromedia Flash, etc. In some embodiments,the system is compatible with Internet Browsers, such as InternetExplorer, Mozilla Firefox, Opera, Safari, etc. In some embodiments, thesystem is compatible with mobile devices having full HTTP/HTTPS support,such as IPHONE, ANDROID, SAMSUNG, POCKETPCs, MICROSOFT SURFACE, videogaming consoles, and the like. Others may include, for example, IPAD andITOUCH devices. In some embodiments, the system can be accessed using aWireless Application Protocol (WAP). This protocol will serve the nonHTTP enabled mobile devices, such as Cell Phones, BLACKBERRY devices,etc., and provides a simple interface. Due to protocol limitations, theFlash animations are disabled and replaced with Text/Graphic menus. Insome embodiments, the system can be accessed using a Simple ObjectAccess Protocol (SOAP) and Extensible Markup Language (XML). By exposingthe data via SOAP and XML, the system provides flexibility for thirdparty and customized applications to query and interact with thesystem's core databases. For example, custom applications could bedeveloped to run natively on APPLE devices, Java or .Net-enabledplatforms, etc. One of skill will appreciate that the system is notlimited to any of the platforms discussed above and will be amenable tonew platforms as they develop.

The web server 620 is typically at least one computer system whichoperates as a server computer system and is configured to operate withthe protocols of the world wide web and is coupled to the internet.Optionally, the web server 620 can be part of an ISP which providesaccess to the internet for client systems. The web server 620 is showncoupled to the server computer system 625 which itself is coupled to webcontent 695, which can be considered a form of a media database. Whiletwo computer systems 620 and 625 are shown in FIG. 6, the web serversystem 620 and the server computer system 625 can be one computer systemhaving different software components providing the web serverfunctionality and the server functionality provided by the servercomputer system 625 which will be described further below.

Cellular network interface 643 provides an interface between a cellularnetwork and corresponding cellular devices 644, 646 and 648 on one side,and network 605 on the other side. Thus cellular devices 644, 646 and648, which may be personal devices including cellular telephones,two-way pagers, personal digital assistants or other similar devices,may connect with network 605 and exchange information such as email,content, or HTTP-formatted data, for example. Cellular network interface643 is coupled to computer 640, which communicates with network 605through modem interface 645. Computer 640 may be a personal computer,server computer or the like, and serves as a gateway. Thus, computer 640may be similar to client computers 650 and 660 or to gateway computer675, for example. Software or content may then be uploaded or downloadedthrough the connection provided by interface 643, computer 640 and modem645.

Client computer systems 630, 650, and 660 can each, with the appropriateweb browsing software, view HTML pages provided by the web server 620.The ISP 610 provides internet connectivity to the client computer system630 through the modem interface 635 which can be considered part of theclient computer system 630. The client computer system can be, forexample, a personal computer system, a network computer, a web TVsystem, or other such computer system.

Similarly, the ISP 615 provides internet connectivity for client systems650 and 660, although as shown in FIG. 6, the connections are not thesame as for more directly connected computer systems. Client computersystems 650 and 660 are part of a LAN coupled through a gateway computer675. While FIG. 6 shows the interfaces 635 and 645 as generically as a“modem,” each of these interfaces can be an analog modem, isdn modem,cable modem, satellite transmission interface (e.g. “direct PC”), orother interfaces for coupling a computer system to other computersystems.

Client computer systems 650 and 660 are coupled to a LAN 670 throughnetwork interfaces 655 and 665, which can be ethernet network or othernetwork interfaces. The LAN 670 is also coupled to a gateway computersystem 675 which can provide firewall and other internet relatedservices for the local area network. This gateway computer system 675 iscoupled to the ISP 615 to provide internet connectivity to the clientcomputer systems 650 and 660. The gateway computer system 675 can be aconventional server computer system. Also, the web server system 620 canbe a conventional server computer system.

Alternatively, a server computer system 680 can be directly coupled tothe LAN 670 through a network interface 685 to provide files 690 andother services to the clients 650, 660, without the need to connect tothe internet through the gateway system 675.

Through the use of such a network, for example, the system can alsoprovide an element of social networking, whereby users can contact otherusers having similar subject-profiles, or user can contact anyone in thepublic to forward the personalized information. In some embodiments, thesystem can include a messaging module operable to deliver notificationsvia email, SMS, TWITTER, FACEBOOK, LINKEDIN, and other mediums. In someembodiments, the system is accessible through a portable, single unitdevice and, in some embodiments, the input device, the graphical userinterface, or both, is provided through a portable, single unit device.In some embodiments, the portable, single unit device is a hand-helddevice.

Regardless of the information presented, the system includes a broaderconcept of a platform for the research community, whether corporate,academic, private, or not-for-profit, for example, to communicate in anengaging way, whether confidential or public. For example, the systemsand methods taught herein can enable researchers to use acomputer/mobile network mobile interface to propose problems andsolutions, offer data, request data, and otherwise communicate regardingissues of common interest. The systems and methods presented herein canbe considered a “game-changer” in art of research and development usingcomputer modeling.

It should be also appreciated that the methods and displays presentedherein, in some embodiments, are not inherently related to anyparticular computer or other apparatus, unless otherwise noted. Variousgeneral purpose systems may be used with programs in accordance with theteachings herein, or it may prove convenient to construct a specializedapparatus to perform the methods of some embodiments. The requiredstructure for a variety of these systems will be apparent to one ofskill given the teachings herein. In addition, the techniques are notdescribed with reference to any particular programming language, andvarious embodiments may thus be implemented using a variety ofprogramming languages. Accordingly, the terms and examples providedabove are illustrative only and not intended to be limiting; and, theterm “embodiment,” as used herein, means an embodiment that serves toillustrate by way of example and not limitation. The following examplesare illustrative of the uses of the present invention. It should beappreciated that the examples are for purposes of illustration and arenot to be construed as limiting to the invention.

Example 1. Pharmacokinetics Modeling

The systems and methods taught herein can be used in pharmacokinetic(PK) models. In this example, a compartmental approach was used in a PKmodel to show the advantages of using the non-mechanistic formulationsand modeling approaches taught herein.

PK models are often used to describe the fate of substances administeredexternally to a living organism. In drug development, they are typicallyused to model the concentration of a drug in the bloodstream after oral,intravenous, or subcutaneous introduction into the body. PK analysis isperformed by non-compartmental or compartmental methods.Non-compartmental methods estimate the exposure to a drug by estimatingparameters such as area under the concentration-time curve (AUC), meanresidence time, clearance, elimination half-life, elimination rateconstant, peak plasma concentration (C_(max)), time to reach C_(max),and minimum inhibitory concentration (MIC). Compartmental methodsestimate the concentration-time graph using kinetic models. Theadvantage of compartmental over some non-compartmental analyses is theability to predict the concentration at any time. The disadvantage isthe difficulty in developing and validating the proper model.

1.1 Compartmental Pharmacokinetics

FIG. 7 shows a prior art, two-compartment linear model that wasconstructed to model the PK behavior of a particular drug, according tosome embodiments. In this example, the first compartment represents thegastro-intestinal (GI) region and the second represents plasma.

The resulting differential equations are:

${V_{i}\frac{\partial C_{i}}{\partial t}} = {{{- k_{f}}C_{i}} + {k_{r}C_{p}}}$${{V_{p}\frac{\partial C_{p}}{\partial t}} = {{k_{f}C_{i}} - {( {k_{r} + k_{e}} )C_{p}}}};$

where, C_(i) and C_(p) are the concentrations of the drug in the GI andplasma compartments, respectively; V_(i) and V_(p) are the volumes ofdistribution for the GI and plasma compartments, respectively; andk_(f), k_(r), and k_(e) are the reaction rate constants. The initialconditions for this model are C_(i)(0)=initial dose=C₀, C_(p)(0)=0.

The species of interest in this example is the plasma concentration,C_(p). The solution to this system of differential equations for C_(p)is:

$\begin{matrix}{{{C_{p}(t)} = {{MC}_{0}( {e^{\beta_{1}t} - e^{\beta_{2}t}} )}};\mspace{14mu}{\beta_{1} > \beta_{2}};} & \; \\{{where},} & \; \\{\beta_{1} = \frac{\frac{- k_{f}}{V_{i}} - \frac{k_{r} + k_{e}}{V_{p}} + \sqrt{( {\frac{k_{f}}{V_{i}} + \frac{k_{r} + k_{e}}{V_{p}}} )^{2} - \frac{4\; k_{f}k_{e}}{V_{i}V_{p}}}}{2}} & (10) \\{\beta_{2} = \frac{\frac{- k_{f}}{V_{i}} - \frac{k_{r} + k_{e}}{V_{p}} + \sqrt{( {\frac{k_{f}}{V_{i}} + \frac{k_{r} + k_{e}}{V_{p}}} )^{2} - \frac{4\; k_{f}k_{e}}{V_{i}V_{p}}}}{2}} & (11) \\{M = \frac{k_{f}}{V_{p}\sqrt{( {\frac{k_{f}}{V_{i}} + \frac{k_{r} + k_{e}}{V_{p}}} )^{2} - \frac{4\; k_{f}k_{e}}{V_{i}V_{p}}}}} & (12)\end{matrix}$

Note that, regardless of the parameter values, the solution for C_(p)(t)is linear with respect to the initial dose; i.e., solutions fordifferent initial doses are simply scalar multiplies of one another.

FIG. 8 shows the data used to calibrate this model (find optimalparameter values), a two-compartment linear model that was constructedto model the PK behavior of a particular drug, according to someembodiments. Doses of 25 mg, 100 mg, and 400 mg were administeredorally. See, for example, Bergman, A., et al. Biopharm. Drug Dispos.,28: 307-313 (2007), which is hereby incorporated herein by reference inits entirety.

When the solution variables β₁, β₂, and M are optimized to yield thebest fit for all of the data, the resulting optimal values are:

$\begin{matrix}{\beta_{1} = {- 0.0025}} \\{\beta_{2} = {- 0.0165}} \\{M = 13}\end{matrix};$

which gives the solution for any initial dose as

C_(p)(t) = 13 C₀(e^(−0.025  t) − e^(−0.0165  t)).

In this case, the optimized solution variables give the best fit for themiddle dose data, while overestimating the lower-dose data andunderestimating the higher-dose data.

FIG. 9 shows a linear two-compartment model solute on for C_(p)(t)compared to data for the pharmacokinetic modeling, according to someembodiments. In particular, the model solution for C_(p)(t) is comparedto the data for each of the 25 mg, 100 mg, and 400 mg cases. It showsthat the model provides a good fit to the 100 mg data, but there is anoverestimation of the 25 mg data and a significant underestimation ofthe 400 mg data.

One limitation of the mechanistic modeling approach—the inability of thelinear two-compartment model to accurately model the fate of the drugover the entire range of dose values; i.e., an insufficiency. Themechanistic approach lacks the necessary structure to adequately modelthe PK of this drug over the entire range of dose values. In this case,an insufficiency is that one of reaction rates is non-linear rather thanlinear. Adding compartments in this case will not improve the results.

Another limitation of the mechanistic modeling approach—ambiguity ofmodel parameters, as shown by the following analysis: Equations(10)-(12) give expressions for solution variables (β₁, β₂, and M) interms of model parameters (k_(f), k_(r), k_(e), V_(i) and V_(p)). Inorder to find the values of model parameters that correspond to a givenset of optimal values for solution variables, we must find expressionsfor model parameters in terms of solution variables. These expressionsare found by enforcing the constraints that all model parameters begreater than zero.

$\begin{matrix}{{- \beta_{1}} \leq \frac{k_{f}}{V_{i}} \leq {- \beta_{2}}} & (13) \\{V_{p} = \frac{k_{f}}{M( {\beta_{1} - \beta_{2}} )}} & (14) \\{k_{e} = \frac{\beta_{1}\beta_{2}V_{i}V_{p}}{kf}} & (15) \\{k_{r} = {{( {{- \beta_{1}} - \beta_{2}} )V_{p}} - \frac{k_{f}V_{p}}{V_{i}} - k_{e}}} & (16)\end{matrix}$

By choosing any combination of k_(f) and V_(i) that satisfies condition(13), one can then solve for the remaining parameters V_(p), k_(e), andk_(r) using the given solution variable values (β₁, β₂, and M) andEquations (14)-(16). Therefore, in this particular example withβ₁=−0.0025, β₂=−0.0165, and M=13, the optimal values for k_(f), V_(i),V_(p), k_(e), and k_(r) are any that satisfy the following conditions:

$\begin{matrix}{0.0025 \leq \frac{k_{f}}{V_{i}} \leq 0.0165} & (17) \\{V_{p} = {5.49*k_{p}}} & (18) \\{k_{e} = \frac{V_{i}V_{p}}{24242*k_{f}}} & (19) \\{k_{r} = {{0.019*V_{p}} - \frac{k_{f}V_{p}}{V_{i}} - k_{e}}} & (20)\end{matrix}$

By choosing any combination of k_(f) and V_(i) that satisfies condition(17), one can then solve for the remaining parameters V_(p), k_(e), andk_(r) using Equations (18)-(20). Thus, while there is only one set ofsolution variable values that result from a given set of model parametervalues (Equations (10)-(12)), there are an infinite number of modelparameter values that can result from a given set of solution variablevalues (Equations (13)-(16)). This non-unique solution toparameter-mapping illustrates the ambiguity that is present in amechanistic approach to modeling, where model parameters are used asintermediaries between inputs and model outputs (solution functions).This ambiguity makes it difficult-to-impossible to map input propertiesto output solutions by way of model parameters.

1.2 Non-Compartmental Pharmacokinetics

The systems and methods taught herein are non-compartmental in design.Non-compartmental PK analysis fits concentration-time curves toavailable data, and then uses these curves to estimate parameters suchas AUC, half-life, C_(max), and time to reach C_(max). The PK parameterscan then be used, for example, to describe the behavior of a drug afterit is introduced into the body.

The systems and methods taught herein are different than traditionalnon-compartmental PK approaches for at least the reason that traditionalapproaches use a mathematical formulation similar to Equation (1), whichdescribes a linear system. The systems and methods taught herein, forexample, are also able to automatically describe non-linearities in thesystem and give more accurate fits to the data. Moreover, there is theproblem of non-unique mappings, which is also an issue with currentnon-compartmental PK analyses. Different concentration-time curves canhave the same AUC but different C_(max), or the same C_(max), butdifferent AUC, for example. And, different concentration-time curves canhave the same AUC but different shapes, resulting in the time aboveminimum concentration being different (different clearance rates). Oneof skill will appreciate that such ambiguities make it difficult to mapproperties of an input compound to its PK parameters, significantlyimpacting the value of PK properties in making predictions of thebehavior of potential drug compounds in a system.

The Systems and Methods Taught Herein Yield Predictions that are MoreAccurate than Current State-of-the-Art Methods

Using the systems and methods taught herein, we can construct a modelthat yields accurate predictions of the fate of the drug over the entirerange of dose values. The systems and methods taught herein provideequation (9), as taught herein for example, which is a three-term modelthat worked well for this particular PK example. Optimization of theresponse variables for the C_(p)(t) function gives the followingoptimized values of the variables in the response function of equation(9) for the PK example:

TABLE 1 K K_(p) α_(p) term i M_(i) ⁰ M_(i) ¹ N_(i) ⁰ N_(i) ¹ 2.283 0.1500.0010 0 −0.028 0.081 — — 1 −3.430 −10.433 0.0040 0.0037 2 4.432 10.1210.0667 0.0274

It should be appreciated that the systems and methods taught hereinprovided a simplified modeling approach, as regardless of how manycompartments or nonlinear reactions might have been attempted to achievesufficient accuracy from a mechanistic approach to this problem, thesystems and methods provided herein were sufficient with only the 13values shown in Table 1.

FIG. 10 shows the C_(p)(t) response function compared to the data foreach of the 25 mg, 100 mg, and 400 mg cases, according to someembodiments. As seen in FIG. 10, the systems and methods taught hereinuse the C_(p)(t) response function to fit the data very well,illustrating that the systems and methods taught herein can accuratelycapture the inherent nonlinearity and, therefore, accurately model thefate of the drug over the entire range of dose values.

The additional degrees of freedom in the systems and methods taughtherein provided a model that was more accurate than the compartmentalmodel. In this example, the mechanistic compartment model contains onlyfive model parameters and therefore involves fewer degrees of freedomthan the systems and methods taught herein. In contrast, the twocompartments, linear reactions, and five parameters in the compartmentalmodel were not sufficient, as they did not adequately model the fate ofthe drug over the entire range of dose values. One of skill willappreciate that such current, state-of-the-art models can easily becomelarge and involve hundreds of parameters. The systems and methods taughtherein, however, provided sufficient accuracy using much fewer degreesof freedom, reducing the ambiguity that is otherwise present in themechanistic approach with its large number of parameters.

Example 2. Enzyme Reaction Modeling (Non-Linear Kinetics)

This example models enzymatic reactions, which are inherently nonlinearin nature. Many input-response models are constructed using anassumption of linear reaction kinetics, which is often insufficient,particularly for large-scale and complex phenomena. One of skill willappreciate that, as shown in the previous PK example, a linear model maynot accurately describe the fate of a drug over a wide range of inputdoses.

Enzyme kinetics is the study of the chemical reactions that arecatalyzed by enzymes. The effects of reaction conditions on reactionrate are investigated which can reveal the catalytic mechanism of theenzyme, its role in metabolism, how its activity is controlled, and howa drug or an agonist might inhibit the activity. Typically, an enzymaticreaction involves an enzyme E binding to a substrate S to form a complexES, which in turn is converted to a product P and the enzyme. This isrepresented schematically as:

${{E + S}\underset{k_{r}}{\overset{k_{f}}{leftharpoons}}{ES}}\overset{k_{cat}}{arrow}{E + P}$

where k_(f), k_(r), and k_(cat) denote the rate constants.

Applying the law of mass action, which states that the rate of areaction is proportional to the product of the concentrations of thereactants, gives a system of four non-linear differential equations thatdefine the rate of change of reactants with time t:

$\begin{matrix}{\frac{\partial\lbrack S\rbrack}{\partial t} = {{- {{k_{f}\lbrack E\rbrack}\lbrack S\rbrack}} + {k_{r}\lbrack{ES}\rbrack}}} & (21) \\{\frac{\partial\lbrack E\rbrack}{\partial t} = {{- {{k_{f}\lbrack E\rbrack}\lbrack S\rbrack}} + {k_{r}\lbrack{ES}\rbrack} + {k_{cat}\lbrack{ES}\rbrack}}} & (22) \\{\frac{\partial\lbrack{ES}\rbrack}{\partial t} = {{{k_{f}\lbrack E\rbrack}\lbrack S\rbrack} - {k_{r}\lbrack{ES}\rbrack} - {k_{cat}\lbrack{ES}\rbrack}}} & (23) \\{\frac{\partial\lbrack P\rbrack}{\partial t} = {k_{cat}\lbrack{ES}\rbrack}} & (24)\end{matrix}$

In this mechanism, the enzyme E is a catalyst, which only facilitatesthe reaction, so its total concentration, free plus combined,[E]+[ES]=[E]₀, is a constant. This conservation law can also be obtainedby adding Equations (22) and (23). This system is nonlinear because ofthe products [E][S] that appear.

If you make the assumption that the concentration of the intermediatecomplex does not change on the time-scale of product formation, then

$\frac{\partial\lbrack{ES}\rbrack}{\partial t} = { 0\Rightarrow{{k_{f}\lbrack E\rbrack}\lbrack S\rbrack}  = {{k_{r}\lbrack{ES}\rbrack} + {{k_{cat}\lbrack{ES}\rbrack}.}}}$

Combining this with the enzyme concentration law gives:

$\lbrack{ES}\rbrack = \frac{\lbrack E\rbrack_{0}\lbrack S\rbrack}{\frac{k_{r} + k_{cat}}{k_{f}} + \lbrack S\rbrack}$

From Equation (24),

$\frac{\partial\lbrack P\rbrack}{\partial t} = {{k_{cat}\lbrack{ES}\rbrack} = \frac{{k_{cat}\lbrack E\rbrack}_{0}\lbrack S\rbrack}{\frac{k_{r} + k_{cat}}{k_{f}} + \lbrack S\rbrack}}$

If we define the following constants,

$\begin{matrix}{V_{\max} = {k_{cat}\lbrack E\rbrack}_{0}} & {{the}\mspace{14mu}{maximum}\mspace{14mu}{reaction}{\mspace{11mu}\;}{velocity}} \\{K_{m} = \frac{k_{r} + k_{cat}}{k_{f}}} & {{the}\mspace{14mu}{Michaelis}\mspace{14mu}{constant}}\end{matrix};$

then, we arrive at the Michaelis-Menten model of enzyme kinetics

$\begin{matrix}{{\frac{\partial\lbrack P\rbrack}{\partial t} = \frac{V_{\max}\lbrack S\rbrack}{K_{m} + \lbrack S\rbrack}},} & (25)\end{matrix}$

which relates the rate of product formation to the concentration ofsubstrate.

State-of-the-Art Michaelis-Menten Models Create Error Due to InvalidAssumptions

Michaelis-Menten-type rates are not only used to model enzyme kineticsbut are also used to model other saturable, nonlinear phenomena.Michaelis-Menten-type rates are often used in mechanistic compartmentmodeling to describe the nonlinear rate at which one species in a systemis produced as a function of the concentration of some other species inthe system. For this reason, it provides a very useful and practicalexample for comparing the systems and methods taught herein to typicalmechanistic approaches to modeling nonlinear phenomena. The problem withusing Michaelis-Menten-type rates for applications other than those forwhich it was derived is that the assumptions used to derive theapproximation might not be applicable. For example, two assumptions usedin deriving the Michaelis-Menten approximation are 1) k_(cat)<<k_(r),and 2) E₀ (the initial enzyme concentration)<<S₀ (the initial substrateconcentration). But these assumptions might not always be valid whenattempting to use a Michaelis-Menten-type rate between two compartmentsin a systems biology model, which is often done.

To illustrate the error involved in using Michaelis-Menten-type rateswhen the underlying assumptions might not be valid, consider a systemwhere k_(f)=k_(f)=k_(cat)=0.2 and E₀=10.0. A Michaelis-Mentenapproximation (Equation (25)) of this system would have V_(max)=2.0 andK_(m)=2.0. It was found that using the systems and methods taughtherein, using Equation (9) in some embodiments, a two-term model wassufficient for this particular enzymatic reaction example. Optimizationof the solution variables for the P(t) function gives optimized valuesof the variables in Equation 9 as shown in Table 2 for the enzymereaction modeling:

TABLE 2 K K_(p) α_(p) term i M_(i) ⁰ M_(i) ¹ N_(i) ⁰ N_(i) ¹ 1.065 2.7980.1986 0 −0.002 −0.038 — — 1 0.992 0.057 0.2352 −0.0866

FIG. 11 shows the P(t) response function compared to data for the enzymereaction modeling, according to some embodiments. As shown in FIG. 11,the solution for P(t) using the system of differential equations(Equations (21)-(24)) can be compared to P(t) obtained using theMichaelis-Menten approximation and to that obtained using the systemsand methods taught herein, for S₀ values of 5.0, 10.0, and 20.0. Thesymbols represent the solution for P(t) from the system of differentialequations, the dashed line represents the solution for P(t) using theMichaelis-Menten approximation, and the solid line represents thesolution for P(t) using the systems and methods taught herein. Thelowest lines are the solutions for the S₀=5.0 case, the middle set oflines are the solutions for the S₀=10.0 case, and the top lines are thesolutions for the S₀=20.0 case. As can be readily seen from FIG. 11, thestate-of-the-art method of using the Michaelis-Menten approximationshows a substantially inferior predictive power than the systems andmethods taught herein.

One of skill will appreciate that the systems and methods taught hereinprovide a much more accurate representation for the solution of thesystem of differential equations over the entire range of initialsubstrate concentrations.

Example 3. Pharmacodynamic Modeling

This example compares the results of a published pharmacodynamic modelto a model constructed using the systems and methods taught herein. Fromthis example, one of skill will appreciate that the systems and methodstaught herein provide a more accurate viral load response predictionthan that obtained using the published, state-of-the-art large-scalecompartmental model which contains many compartments, differentialequations, nonlinear reactions, and parameters.

While PK models are used to describe the fate of substances administeredexternally to a living organism, pharmacodynamic (PD) models are used todescribe the response of some system entity to the introduction of asubstance administered externally. It is often said that PK modelsdescribe what the body does to a drug, whereas PD models describe whatthe drug does to the body. In terms of input-response, PK modelsdescribe the response of the input compound upon introduction into thebody, while PD models describe the response of some other system entityafter introduction of a certain compound. Both are input-responsemodels, but in PD modeling, the response of interest is a systemcomponent that is different than the input compound. For example, a PDmodel might describe the amount of a certain type of infectious bacteriathat is present over time after introduction of a specific antibiotic;whereas, a PK model would describe the fate of the antibiotic over time.

The published model is a PD model designed to predict HIV viral loadresponse to the administration of the drug tenofovir in oral doses of75, 150, 300, and 600 mg. See Duwal, S., et al. PLoS One, 7(7):e40382(2012), which is hereby incorporated herein by reference in itsentirety. The published PD model is coupled to a pharmacokinetic model,a four-compartment model containing both linear and nonlinearMichaelis-Menten kinetics, and it consists of a nonlinear system ofeight differential equations. As such, the coupledpharmacokinetic-pharmacodynamic model is a mechanistic model containing12 species and 31 free parameters. As a virus dynamics model, it wasused to predict viral loads following tenofovir treatment inHIV-infected patients.

FIGS. 12A and 12B illustrate the pharmacokinetic and pharmacodynamicmodel as used in predicting viral loads in response to administration oftenofovir, according to some embodiments. FIG. 12A is a drawing of apharmacokinetic model of the system, and FIG. 12B is a drawing of avirus dynamics model. In FIG. 12A, D refers to an input dose oftenofovir disoproxil fumurate (TDF), an antiviral pro-drug, in asubject. With respect to plasma PK, C₁ is a compartment that resemblesplasma pharmacokinetics, and C₂ is a compartment for the poorly perfused(peripheral) tissues in the pharmacokinetic model. With respect to cellPK, C_(cell) resembles the concentrations of tenofovir disphosphate(TFV-DP) in peripheral blood mononuclear cells. Parameters k₁₂ and k₂₁are the rate constants for influx and outflux to/from the peripheralcompartment C₂, and k_(a) and k_(e) are the rates of TFV uptake for theelimination into/out-of C₁, respectively. F_(bio) is bioavailability.V_(max) and k_(m) are Michaelis-Menten kinetics parameters, and k_(out)is the cellular elimination rate constant of TFV-DP. See, for example,pages 2 and 3 of Duwal, S., et al. PLoS One, 7(7):e40382 (2012).

FIG. 12B is coupled to FIG. 12A in that the β_(T), β_(M), CL_(T), andCL_(M) parameters in the pharmacodynamics model are functions of theC_(cell) concentration from the pharmacokinetics model. In FIG. 12B, Inbrief, the virus dynamics model comprises T-cells, macrophages, freenon-infectious virus (T_(U),M_(U),V_(NI), respectively), free infectiousvirus V₁, and four types of infected cells: infected T-cells andmacrophages prior to proviral genomic integration (T₁ and M₁,respectively) and infected T-cells and macrophages after proviralgenomic integration (T₂ and M₂, respectively). λ_(T) and λ_(M) are thebirth rates of uninfected T-cells and macrophages, and δ_(T) and δ_(M)denote their death rate constants. The parameters δ_(PIC,T) andδ_(PIC,M) refer to the intracellular degradation of essential componentsof the pre-integration complex, e.g., by the host cell proteasome, whichreturn early infected T-cells and macrophages to an uninfected stage,respectively. Parameters β_(T) and β_(M) denote the rate of successfulvirus infection of T-cells and macrophages in the presence of TFV-DP,respectively, while the parameters CL_(T) and CL_(M) denote theclearance of virus through unsuccessful infection of T-cells andmacrophages in the presence of TFV-DP. Parameters k_(T) and k_(m) arethe rate constants of proviral integration into the host cell's genomeand N_(T)* and N_(M)* denote the total number of released infectious andnon-infectious virus from late infected T-cells and macrophages andN_(T) and N_(M) are the rates of release of infectious virus. Theparameters δ_(T1), δ_(T2), δ_(M1) and δ_(M2) are the death rateconstants of T₁, T₂, M₁, and M₂ cells, respectively. The free virus(infectious and non-infectious) gets cleared by the immune system withthe rate constant CL. See, for example, pages 3 and 4 of Duwal, S., etal. PLoS One, 7(7):e40382 (2012).

The complicated modeling shown by FIGS. 12A and 12B can be simplifiedusing the systems and methods taught herein. It was found, for example,that using the systems and methods taught herein, which includes usingEquation (9), a three-term model was sufficient. Optimization of theresponse variables for the viral load function gives the followingoptimized values of the variables as shown in Table 3:

TABLE 3 K K_(p) α_(p) term i M_(i) ⁰ M_(i) ¹ N_(i) ⁰ N_(i) ¹ 0.233 5.0100.0211 0 0.76 0.14 — — 1 26.21 7.63 0.0007 0.0054 2 −5.24 2.68 −0.00040.0131

FIG. 13 shows a plot of the responses provided using the systems andmethods taught herein as compared to the large-scale compartment model,according to some embodiments. The published model (PM) was taken fromDuwal, et al. See Duwal, S., et al. PLoS One, 7(7):e40382 (2012), asdescribed herein. The systems and methods taught herein are the newmodel (NM) and are compared to PM. Dashed and dotted lines represent PM,the predicted median viral kinetics, using the model of Duwal. Thesymbols represent actual data points from the observed viral kinetics,and the solid lines represent predicted responses using the systems andmethods taught herein, NM. Once daily 75 mg TDF dosing and once daily300 mg TDF dosing are shown.

As shown in FIG. 13, the new model is able to accurately capture thesame input-response behavior that is produced by the larger mechanisticmodel. This increased level of accuracy is important not only in dosingstudies of tenofovir but also in creating more accurate predictions ofviral load response to test input compounds other than tenofovir.Surprisingly, the systems and methods taught herein functioned very wellwith only 13 response variables rather than the 31 model parameters usedby the state-of-the-art model. As such, the systems and methods taughtherein are less prone to the ambiguity in model parameter to solutionmapping that is present in a mechanistic model. One of skill willappreciate this surprising and unexpected control over such ambiguities,particularly if one were to try to make predictions of thepharmacodynamic response based on properties of input compounds.

Example 4. Quantitative Structure-Activity Relationship (QSAR)Predictions

This example shows that the systems and methods taught herein can beused to determine quantitative structure-activity relationships (QSAR),the mapping of molecular structure properties of an input compound to aresponse, or activity, within a given system. QSAR allows one of skill,for example, to (i) summarize a relationship between chemical structuresand biological activity in a dataset of chemicals; and (ii) predict theactivities of new chemicals. It is this same type of characterizationand prediction that can be obtained with the systems and methods taughtherein, significantly impacting a wide variety of fields, including drugdesign and personalized medicine. One of skill will appreciate that thesystems and methods taught herein can be used to relate properties of aninput to a particular response profile and address the desire to relatethe variables of an input-response model (the model parameters in amechanistic model or the response function variables in the systems andmethods taught herein, for example) to properties of the input.Moreover, one of skill will also appreciate the systems and methodstaught herein for their ability to relate parameters of a dose responsemodel in drug design to the molecular properties of a proposed drug(input compound). The accurate mapping of input molecular properties tomodel parameters allows the art to input compounds covering a wide rangeof molecular properties and get an accurate description of the resultingresponse for each. Accordingly, the systems and methods taught hereinprovide the basis for an ‘in silico’ screening process, where one couldselect an input compound that yields the most desirable response.

Mechanistic models lack the necessary one-to-one relationships betweenmodel parameters and model output. As demonstrated in previous examples,this is why such mechanistic models are often unable to producesufficient maps of input properties to model parameters. This is aproblem of “a lack of specificity,” in that it is possible to achievethe same output from many different sets of model parameters.Unfortunately, this lack of specificity between parameters and output isa serious problem in that it becomes impossible to expose uniqueinput-response relationships. For example, by way of ambiguousparameters, the same input could produce a wide range of responses, ormany different inputs could produce the same response. The systems andmethods taught herein, however, can reduce or even eliminate thisambiguity, and allow for more accurate mappings between input propertiesand output (response) profiles via the response function variables.

Using Molecular Properties to Select Drug Candidates

Molecular properties are often used to determine if a chemical compoundwith a certain pharmacological or biological activity has propertiesthat would make it a likely orally active drug in humans. Suchproperties can include, but are not limited to, number of hydrogen bonddonors, number of hydrogen bond acceptors, molecular weight,octanol-water partition coefficient, electrostatic potential, surfacecharge, surface potential, density, ionization energy, H_(vaporization),H_(hydration), lipophilicity parameter, pK_(a), boiling point,refractive index, dipole moment, reduction potential, ovality, HOMOenergy, polarizability, molecular volume, vdW surface area, molecularrefractivity, hydration energy, surface area, LUMO energy, charges onindividual atoms, solvent accessible surface area, maximum + and −charge, hardness, Taft's steric parameter, 3D configuration of atoms,and secondary structure such as helices, beta strands, beta sheets,coils, and loops. Molecular properties that are more geometrical innature are used, for example, to determine if a chemical compound meetsthe essential, or desired, structural parameters for binding with areceptor. Because the systems and methods taught herein can remove muchof the ambiguity between input properties and response profiles, theywill be more likely to make accurate mappings from biological activityand structural properties of candidate drug molecules to responseprofiles. As such, the systems and methods taught herein can provide anextremely valuable tool for pre-clinical modeling and prediction ofactivity against a given target, or PK-ADME (absorption, distribution,metabolism, and excretion) properties of candidate drug compounds.

The Problem of Ambiguity in Current, State-of-the-Art Modeling

To demonstrate the ambiguity that would arise in attempting to mapmolecular properties of an input compound to variables in the responsefunction, consider the pharmacokinetic modeling problem presented inExample 1. As was shown in that example, there were an infinite numberof model parameter values (k_(f), k_(r), k_(e), V_(i), and V_(p)) thatcould yield the desired values for the variables β₁, β₂, and M in thesolution function C_(p)(t) when using a linear, mechanistic,compartmental modeling approach. A typical QSAR study of this problemwould attempt to map molecular properties of an input compound to modelparameter values. For example, if molecular weight (W) and partitioncoefficient (log P) were the predominant factors in the pharmacokineticproperties of a compound, then one would attempt to describe the modelparameters k_(f), k_(r), and k_(e) as functions of Wand log P (it isassumed that V_(i), and V_(p) are parameter values that would have to beestimated but would be independent of Wand log P). Once such functionsare constructed, the values of the response function variables (β₁, β₂,and M) would be directly determined by the molecular weight andpartition coefficient of the input compound. This is shownmathematically below:

$\begin{matrix} {\begin{matrix}{\beta_{1} = {F_{1}( {k_{f},k_{r},k_{e}} )}} \\{\beta_{2} = {F_{2}( {k_{f},k_{r},k_{e}} )}} \\{M = {F_{3}( {k_{f},k_{r},k_{e}} )}}\end{matrix}\mspace{31mu}\begin{matrix}{k_{f} = {G_{1}( {W,{\log\; P}} )}} \\{k_{r} = {G_{2}( {W,{\log\; P}} )}} \\{k_{e} = {G_{3}( {W,{\log\; P}} )}}\end{matrix}}\Rightarrow\begin{matrix}{\beta_{1} = {{H_{1}( {W,{\log\; P}} )} = {F_{1}( {{G_{1}( {W,{\log\; P}} )},{G_{2}( {W,{\log\; P}} )},{G_{3}( {W,{\log\; P}} )}} )}}} \\{\beta_{2} = {{H_{2}( {W,{\log\; P}} )} = {F_{2}( {{G_{1}( {W,{\log\; P}} )},{G_{2}( {W,{\log\; P}} )},{G_{3}( {W,{\log\; P}} )}} )}}} \\{M = {{H_{3}( {W,{\log\; P}} )} = {F_{3}( {{G_{1}( {W,{\log\; P}} )},{G_{2}( {W,{\log\; P}} )},{G_{3}( {W,{\log\; P}} )}} )}}}\end{matrix}  & (26)\end{matrix}$

Therefore, given the molecular weight and partition coefficient of aninput compound, the values of response function variables could becomputed directly, thus giving a complete time-course pharmacokineticprofile of that compound. Examples of F₁, F₂, and F₃ functions weregiven in Example 1, Equations (10)-(12). Attempting to compute accuratemolecular property to model parameter functions (G₁, G₂, and G₃functions) demands a set of input-response data for input compoundscovering a range of molecular properties. This data would be used tofind the optimal function types and function values for the molecularproperty to model parameter functions.

The limitation of this approach comes from the ambiguity that is presentin attempting to construct the molecular property to model parameterfunctions. For the sake of simplicity, consider the case where molecularweight is the only property that affects response. And consider the samepharmacokinetic problem from Example 1, where the values β₁=−0.0025,β₂=−0.0165, and M=13 were found to provide the best fit to the givenobservations of response (the data sets of responses to given inputs).In that example, expressions were derived for model parameter values asfunctions of solution variable values (Equations (13)-(16)). Theseexpressions showed that for a given set of β₁, β₂, and M values, thereare an infinite number of model parameter values that can result. Theseexpressions also provided bounds for the model parameter values. Thus,the molecular property to model parameter functions must be bounded inthis case. There are many types of functions that can provide suchbounds, but consider the functional form given in Equation (2) usingonly two terms:

${G(W)} = {M^{0} + {M^{1}( \frac{1 - e^{{- \sigma}\; W}}{1 + {ce}^{{- \sigma}\; W}} )}}$

This function is bounded by M⁰+M¹ and M⁰−M¹/c. Using this form to definethe parameter values as functions of W gives:

$k_{f} = {{G_{1}(W)} = {M_{1}^{0} + {M_{1}^{1}( \frac{1 - e^{{- \sigma_{1}}W}}{1 + {c_{1}e^{{- \sigma_{1}}W}}} )}}}$$k_{r} = {{G_{2}(W)} = {M_{2}^{0} + {M_{2}^{1}( \frac{1 - e^{{- \sigma_{2}}W}}{1 + {c_{2}e^{{- \sigma_{2}}W}}} )}}}$$k_{e} = {{G_{3}(W)} = {M_{3}^{0} + {M_{3}^{1}( \frac{1 - e^{{- \sigma_{3}}W}}{1 + {c_{3}e^{{- \sigma_{3}}W}}} )}}}$

Equation (13) from Example 1 gives the allowable range for k_(f) as afunction of the given β₁ and β₂, and the calculated V_(i).

$ {{- \beta_{1}} \leq \frac{k_{f}}{V_{i}} \leq {- \beta_{2}}}\Rightarrow{{- \beta_{1}} \leq \frac{G_{1}(W)}{V_{i}} \leq {- \beta_{2}}} $

Since G₁(W)/V_(i) is bounded by (1/V_(i))(M₁ ⁰+M₁ ¹) and (1/V_(i))(M₁⁰−M₁ ¹/c₁), then

$ {{- \beta_{1}} \leq {\frac{1}{V_{i}}( {M_{1}^{0} + M_{1}^{1}} )} \leq {{{- \beta_{2}}\mspace{14mu}{and}}\mspace{14mu} - \beta_{1}} \leq {\frac{1}{V_{i}}( {M_{1}^{0} - \frac{M_{1}^{1}}{c_{1}}} )} \leq {- \beta_{2}}}\Rightarrow{{{M_{1}^{1}}( {1 + \frac{1}{c_{1}}} )} < {{( {\beta_{1} - \beta_{2}} )V_{i}} - {\beta_{1}V_{i}} - M_{1}^{1}} \leq M_{1}^{0} \leq {{{- \beta_{2}}V_{i}} + {\frac{M_{1}^{1}}{c_{1}}\mspace{14mu}( {{{if}\mspace{14mu} M_{1}^{1}} < 0} )} - {\beta_{1}V_{i}} + \frac{M_{1}^{1}}{c_{1}}} \leq M_{1}^{0} \leq {{{- \beta_{2}}V_{i}} - {M_{1}^{1}\mspace{14mu}( {{{if}\mspace{14mu} M_{1}^{1}} > 0} )}}} ;$

-   Where, β₁<0, β₂<0, V_(i)>0, M₁ ⁰>0, and c₁>0. There are no    constraints placed on σ₁ (i.e., −∞≤σ₁≤∞).

Thus, the allowable values for M₁ ⁰, M₁ ¹, and c₁, are given by the β₁,β₂, and V_(i) values obtained from fitting the data. Thus, there are aninfinite number of values for the variables (M₁ ⁰, M₁ ¹, and c₁) thatdescribes the relationship between the molecular property Wand the modelparameter k_(f). This will also be true of the variables describing therelationship between the molecular property Wand the model parametersk_(r) and k_(e). Depending on the values of β₁, β₂, and V_(i), the rangeof allowable values for M₁ ⁰, M₁ ¹, and c₁ could be quite large.

There are, of course, other types of nonlinear functional forms thatcould be used for the G(W) functions, but all will introduce additionalparameters and the same type of ambiguity will result. Therefore, thenon-unique mappings that exist between model parameters and responsefunctions in a mechanistic model will extend to the mappings betweeninput molecular properties and model parameters in a QSAR study. Thiswill result in a non-unique mapping between input molecular propertiesand output response functions. Such a non-unique mapping will make itprohibitively difficult to obtain accurate and effective QSARpredictions.

Using the Systems and Methods Taught Herein; Eliminating MechanisticModeling Parameters to Reduce Ambiguity

The approach for QSAR prediction using the systems and methods taughtherein is to start with input-response data for input compounds having awide range of molecular properties. For each compound, various doseswould be tested and a model can be built using the new formulation;i.e., optimal values would be found for the response function variablesK, K_(p), α_(p), M₀ ⁰, . . . , M_(n) ⁰, M₀ ¹, . . . , M_(n) ¹, N₁ ⁰, . .. , N_(n) ⁰, and N₁ ¹, . . . , N_(n) ¹. A mapping can then beconstructed between the molecular properties of the input compounds andthe optimal values of the response function variables. Once thismapping, or set of functions, is found, then predictions can be made asto what type of response will result from introduction of a givencompound into the system. All that would be required as input is thespecific values of the molecular properties of a compound. These valueswould then uniquely determine the values of the response functionvariables in the systems and methods taught herein, which would give atime-course profile of the desired response. Using that time-courseprofile, one could evaluate the effectiveness of the input compound inachieving a desired response. The mapping from molecular properties toresponse functions will contain less ambiguity because it eliminates theintermediate step of mechanistic model parameters. It will be much morelikely to obtain accurate mappings between input molecular propertiesand response function variables because of the reduction in ambiguity.With such mappings, virtual screenings can be performed to assess thelikelihood that a particular input compound will produce a desiredresponse. A mathematical description of the QSAR process using thesystems and methods taught herein is given below.

Using available data, models would be set up for each input compoundbased on the observed responses due to various doses. Each of thesemodels would contain optimal values of the response function variablesK, K_(p), α_(p), M₀ ⁰, . . . , M_(n) ⁰, M₀ ¹, . . . , M_(n) ¹, N₁ ⁰, . .. , N_(n) ⁰, and N₁ ¹, . . . , N_(n) ¹. From these models, functionswould be fit that map molecular properties of the input to the variablesin the response function. These functions are analogous to the Hfunctions that were composed in the mechanistic case (Equations (26)).For example, if molecular weight, W, and partition coefficient log Pwere the only molecular properties considered, there would be 4n+5functions, where n is the number of the final term in the responsefunction. Using the optimal values of response function variables thatwere derived for each input compound, and the molecular weight andpartition coefficient of each compound, the following functions(mappings) would be estimated:

$\begin{matrix}{K = {H_{1}( {W,{\log\; P}} )}} \\{K_{p} = {H_{2}( {W,{\log\; P}} )}} \\{\alpha_{p} = {H_{3}( {W,{\log\; P}} )}} \\{M_{0}^{0} = {H_{4}( {W,{\log\; P}} )}} \\\vdots \\{M_{n}^{0} = {H_{n + 4}( {W,{\log\; P}} )}} \\{M_{0}^{1} = {H_{n + 5}( {W,{\log\; P}} )}} \\\vdots \\{M_{n}^{1} = {H_{{2n} + 5}( {W,{\log\; P}} )}}\end{matrix}\mspace{31mu}\begin{matrix}{N_{1}^{0} = {H_{{2n} + 6}( {W,{\log\; P}} )}} \\\vdots \\{N_{n}^{0} = {H_{{3n} + 6}( {W,{\log\; P}} )}} \\{N_{1}^{1} = {H_{{3n} + 6}( {W,{\log\; P}} )}} \\\vdots \\{N_{n}^{1} = {H_{{4n} + 5}( {W,{\log\; P}} )}}\end{matrix}$

In some embodiments, there would typically be more than two molecularproperties considered, and thus the construction of the H functionswould require higher-dimensional approximations. The extension to higherdimensions does not significantly alter the basic approach, but it wouldrequire additional computational cost.

Once the H functions are established, a direct connection is made frommolecular weight and partition coefficient of a compound to values ofthe response function variables. Based on this connection, when a newcompound is considered, its molecular weight and partition coefficientare used to calculate values of the variables in the response function.After calculating the values of the variables in the response function,the result is a full time-course profile of the response. This profilecan then be used to assess properties such as maximum concentration,time to maximum concentration, time above a minimum concentration,clearance, permeability, size of solid tumor, etc.—all of which are veryvaluable in systems biology and drug design modeling. These predictionsof response provide an extremely valuable tool by which large numbers ofcompounds can be screened very quickly using high-speed andlarge-storage computers.

Example 5. Micro-Dosing Studies

Micro-dosing is a technique for studying the behavior of drugs in humansthrough the administration of doses so low (“sub-therapeutic”) that theyare unlikely to produce whole-body effects, but high enough to allow thecellular response to be studied. This allows us to see the PK of thedrug with almost no risk of side effects. This is usually conductedbefore clinical Phase I trials to predict whether a drug is viable forthat phase of testing. Human micro-dosing aims to reduce the resourcesspent on non-viable drugs and the amount of testing done on animals. Asonly micro-dose levels of the drug are used, analytical methods arelimited and extreme sensitivity is needed. Accelerator mass spectrometry(AMS) is the most common method for micro-dose analysis. Many of thelargest pharmaceutical companies have now used micro-dosing in drugdevelopment, and the use of the technique has been provisionallyendorsed by both the European Medicines Agency and the Food and DrugAdministration. It is expected that human micro-dosing will gain asecure foothold at the discovery-preclinical interface driven by earlymeasurement of candidate drug behavior in humans.

There are many reasons for potential drug candidates to be dropped fromthe pharmaceutical pipeline. A suitable compound must demonstrateefficacy in the target patient population and have an acceptable safetyprofile, requirements which are themselves extremely demanding. Oneproperty of a compound that influences these and other factors is its PKprofile. That is, how efficiently the compound is absorbed from the siteof administration into the body, how well it is distributed to varioussites within the body, including the site of action, and how rapidly andby what mechanism(s) it is eliminated, by excretion and metabolism(ADME—absorption, distribution, metabolism and excretion). Furthermore,the vast majority of compounds are metabolized, therefore the fate ofthe newly formed metabolites must be taken into account, as many ofthese are active and some have adverse side effects. It has beenestimated that between 10% and 40% of potential drugs fail during earlyclinical trials because of unsuitable PK features. A poor PK profile mayrender a compound of so little therapeutic value as to be not worthdeveloping. For example, very rapid elimination of a drug from the bodywould make it impractical to maintain a compound at a suitable level tohave the desired effect. Clearly, the ideal is to only test in humansthose compounds that have desirable PK properties. However, this is notrivial task. The problem is that despite significant progress to dategenerally, we are still unable to predict the PK profile in humans ofmany drug classes from in vitro and computer-based methods. We aretherefore reliant on information gained in animals, which is based onpast experience and has been the most predictive, to help screen thecompounds for those with an appropriate PK profile. One commonly-appliedapproach to predicting a human PK profile based on animal data isallometric scaling, which scales the animal data to humans, assumingthat the only difference among animals and humans is body size. Whilebody size is an important determinant of PK, it is certainly not theonly feature that distinguishes humans from animals and, therefore, thissimple approach has been estimated to have less than 60% predictiveaccuracy.

This is where micro-dosing comes in. Clinical testing phases 1 to 3involve evaluating pharmacological doses generally first in humanvolunteers and then in patients for efficacy and safety. The hypothesisis that micro-dosing will help reduce or replace the extensive testingin animals of the many compounds that do not have desirable PKproperties in humans and subsequently would be rejected. But what is amicro-dose, and how could it help? A micro-dose is so small that it isnot intended to produce any pharmacologic effect when administered tohumans and therefore is also unlikely to cause an adverse reaction. Forpractical purposes this dose is defined as 1/100th of that anticipatedto produce a pharmacological effect, or 100 micrograms, whichever is thesmaller. The interest in giving such a micro-dose to humans early in thedrug development process is centered on the view that many of theprocesses controlling the PK profile of a compound are independent ofdose level. Therefore, a micro-dose will provide sufficiently useful PKinformation to help decide whether it is worth continuing compounddevelopment, which includes, for example, toxicity testing in animals.

Computer models can provide valuable analytical tools in the area ofmicro-dosing, although there are serious practical hurdles that must beresolved. As we have seen in the previous examples, a computer modelthat does not accurately capture all of the linear and nonlinear effectswithin a system will not yield accurate extrapolations of low-doseresults to higher-doses. This is where the systems and methods hereinwill have significant positive impact, where in some embodiments theywill include a dose-response model using several different micro-doses,and then extrapolate that model to higher, therapeutic doses.

Testing of the systems and methods taught herein has shown that in truemicro-dosing studies, if the low-dose data used to construct the modelcovers a wide enough range, then accurate predictions can be made fordoses that are roughly one order of magnitude higher. To illustrate thispoint, consider the case of intestinal drug absorption. The absorptionof drugs via the oral route is a subject of intense and continuousinvestigation in the pharmaceutical industry since good bioavailabilityimplies that the drug is able to reach the systemic circulation bymouth. The intestine is an important tissue that regulates the extent ofabsorption of orally administered drugs, since the intestine is involvedin first-pass removal. A simple model of intestinal drug absorptionfocuses on the permeation of a drug compound across the epithelial cellsthat separate the blood vessels and intestines. The ability of acompound to permeate the cell layer is governed by diffusive processesas well as cell membrane transporters that can actively move compoundsin the opposite direction of a concentration gradient. Thesetransporters counteract the permeation of a compound that would occur bydiffusion alone, due to a concentration gradient.

The simple model of intestinal drug absorption can be represented as athree-compartment model, where one compartment represents the intestine,one the cell layer, and the other the bloodstream. Forward and reversediffusion rates can be set up between the compartments, and the cellmembrane transporters can be represented by a non-reversible ratebetween the cell and the intestine. Because the capacity of the cellmembrane transporters is limited, it is a “saturable” process. That is,once the transporters have become saturated with a particular compound,they can no longer accept any more and will then continue to transportat a constant rate. This type of saturable process is nonlinear and istypically modeled using Michaelis-Menten kinetics. The compartment modeland associated differential equations are given below.

FIG. 14 shows a three-compartment model that is used as a simplerepresentation for the absorption of a compound between the intestinesand bloodstream for a dosing study, according to some embodiments. Thecompartment modeling can include the following equations:

${V_{i}\frac{\partial C_{i}}{\partial t}} = {{{- k_{1}}C_{i}} + {k_{2}C_{e}} + {( \frac{V_{m}}{k_{m} + C_{e}} )C_{e}}}$${V_{e}\frac{\partial C_{e}}{\partial t}} = {{k_{1}C_{i}} - {( {k_{2} + k_{3}} )C_{e}} + {k_{4}C_{b}} - {( \frac{V_{m}}{k_{m} + C_{e}} )C_{e}}}$${{V_{b}\frac{\partial C_{b}}{\partial t}} = {{k_{3}C_{e}} - {k_{4}C_{b}}}};$

where, V_(i), V_(e), and V_(b) represent the volumes of distribution forthe intestinal, epithelial cell, and bloodstream compartments,respectively; k₁, k₂, k₃, and k₄ represent the diffusion rates; andk_(m), V_(m) are the Michaelis-Menten rate constants for the activetransport. For the purpose of this example, V_(i)=V_(e)=V_(b)=1.0,k₁=k₂=1.0, k₃=k₄=5.0, k_(m)=1.0, and V_(m)=5.0. The initialconcentrations are all 0 except for the intestinal compartment whoseinitial condition is equal to the input dose, C₀.

In order to perform a dosing study, C₀ values of 1, 10, and 100 mg wereused to construct a model of the absorption of a compound between theintestines and bloodstream using the new formulation and a linear modelthat does not take into account the nonlinear transport effect. It wouldbe reasonable to expect that, given these initial values, a linear modelmight be chosen since that would provide a fairly accurate fit to thedata. The two models were then used to predict the concentration profilein the blood compartment that results from an input dose of 1000 (oneorder of magnitude higher than the highest dose used to construct themodel). The model results were then compared to the numerical solutionof the system of differential equations (referred to as the “data”). Itwas found that using the systems and methods taught herein (usingEquation (9)), a three-term model was sufficient for this particularexample. Optimization of the response variables for the C_(b)(t)function gives the following optimized values of the variables in theresponse function used by the systems and methods taught herein for thedosing study, as shown in Table 4:

TABLE 4 K K_(p) α_(p) term i M_(i) ⁰ M_(i) ¹ N_(i) ⁰ N_(i) ¹ 1.684 1.2590.1225 0 0.0006 −0.0049 — — 1 0.0211 0.1927 2.0551 −0.0427 2 0.1049−0.0003 5.5767 0.0497

FIG. 15 shows the prediction of the bloodstream concentration vs. timeprofile for a 1000 mg dose, using both the linear and systems andmethods taught herein, according to some embodiments. Both (i) thelinear model and (ii) the model of the systems and methods taught hereinare compared to the ‘data,’ or numerical solution. Both the linear modeland the systems and methods taught herein provide accurate fits to theC₀=1, 10, and 100 mg data sets (discussed as observed, but not plotted,for purposes of clarity). But, when you consider the use of the model topredict the C₀=1000 mg data set, FIG. 15 shows that the systems andmethods taught herein provide a significantly more accurate fit to thedata. This is because the systems and methods taught herein were able topick up the nonlinear behavior due to the saturable membrane transportphenomena. It could be argued that one could adjust the mechanisticmodel to reflect the nonlinearity, but it may not be known a prioriwhere the nonlinear phenomena occurs and precisely what the nonlinearkinetic rate(s) should be. The systems and methods taught herein pick upthe nonlinearity automatically and are able to extend that to makeaccurate predictions of response due to higher-dose initial conditions.

Example 6. The Use of Surrogates in Modeling: Biomarkers andMetabolomics

This example shows how the use of surrogates for response data inmodeling to predict a response. Surrogates can include, for example,biomarkers and metabolomics. If the generation of response data isprohibitively expensive or time-consuming, for example, then the use ofbiomarkers or metabolites allows for the construction of a model thatmight otherwise be impossible to build. For example, if the response ofinterest is the size of a solid tumor and we would like to haveobservations over a relatively short time scale (minutes-hours), then wewould have to obtain images of the tumor every few minutes or hours, andthe cost of imaging technology in itself could be prohibitive.

In some embodiments, the term “biomarker” can be used to refer abiological molecule found in blood, other body fluids, or tissues thatis (i) a sign of a normal or abnormal process, or of a condition ordisease; or, (ii) used to see how well the body responds to a treatmentfor a disease or condition. In some embodiments, A biomarker can also becalled “a molecular marker” or “a signature molecule.” In someembodiments, a biomarker can be diagnostic, for example, to helpdiagnose a cancer, perhaps before it is detectable by conventionalmethods. In some embodiments, a biomarker can be prognostic, forexample, to forecast how aggressive the disease process is and/or how apatient can expect to fare in the absence of therapy. And, in someembodiments, a biomarker can be predictive, for example, to helpidentify which patients will respond to which drugs. For example,biomarker can be used as a surrogate indication of the progression of atumor, for example, the measurement of which can be less time-consumingand costly than the measurement of the tumor size. The prostate-specificantigen (PSA) is an example of a protein produced by cells of theprostate gland that can be measured in blood samples, as prostate cancercan increase PSA levels in the blood, making PSA a biomarker forprostate tumors. Other examples of biomarkers include, but are notlimited to, C reactive protein (CRP) for inflammation; high cholesterolfor cardiovascular disease; S100 protein for melanoma; HER-2/neu genefor breast cancer; BRCA genes for breast and ovarian cancers (BRCA1 andBRCA2); CA-125 for ovarian cancer; BNP in heart failure, CEA incolorectal cancer; creatine levels in renal failure; cerebral blood flowfor Alzheimer's disease, stroke, and schizophrenia; high bodytemperature for infection; and, the size of brain structures forHuntington's disease.

Metabolomics uses metabolites as the intermediates and products ofmetabolism, and metabolomics can be used in input-response modeling, forexample, in the area of drug toxicity assessment. In some embodiments,metabolic profiling of a body fluid can be used as a surrogate. In someembodiments, metabolic profiling of urine or blood plasma can be used asa surrogate, for example, to detect the physiological changes caused bytoxic insult of a chemical. Pharmaceutical companies can usemetabolomics in modeling, for example, to test the toxicity of potentialdrug candidates: if a compound can be eliminated before it reachesclinical trials on the grounds of adverse toxicity, it saves theenormous expense of the trials. In some embodiments, the metabolite thatis profiled can be an endogenous metabolite produced by the subject, anexogenous metabolite, or a xenometabolite produced by a foreignsubstance such as a drug. In some embodiments, a metabolite can include,but are not limited to, a lipoprotein or albumin.

In some embodiments, phenyalanine and tyrosine concentrations can beused for diagnosing inborn errors of metabolism (IEM), as they areconsidered as potentially the most clinically applicable metabolicbiomarkers in combination with glucose for diabetes diagnosis.

In some embodiments, metabolites can be used in cancer studies. Forexample, a subset of six metabolites (sarcosine, uracil, kynurenine,glycerol-3-phosphate, leucine and proline) have shown to besignificantly elevated upon disease progression from benign toclinically localized prostate cancer and metastatic prostate cancer. Onemetabolite, sarcosine, has been identified as a potential candidate forfuture development in biomarker panels for early disease detection andaggressivity prediction in prostate cancer. Components of a mammaliansystem that can be used in such studies include, for example, plasma,tissue and urine. Blood serum can be used, for example, as the componentin studies of renal cancer colorectal cancer, pancreatic cancer,leukemia, ovarian cancer, and oral cancer. Urine can be used, forexample, as the component in studies of breast cancer, ovarian cancer,cervical cancer, hepatocellular carcinoma, and bladder cancer. And,saliva can be used, for example, as the component in studies of oralcancer, pancreatic cancer, and breast cancer, as well as periodontaldisease.

In some embodiments, metabolites can be used in cardiovascular studies.For example, pseudouridine, citric acid, and the tricarboxylic acidcycle intermediate 2-oxoglutarate can be used in some embodiments asserum biomarkers. Cardiovascular conditions can include myocardialischemia and coronary artery disease. In some embodiments,dicarboxylacylcarnitines can be used to predict death/myocardialinfarction outcomes. And, in some embodiments, plasma levels ofasymmetric dimethylarginine can be used to predict major adverse cardiacevents in patients with acute decompensated heart failure and withchronic heart failure.

All of the previous examples—PK modeling (Example 1), enzyme reactionmodeling (Example 2), PD modeling (Example 3), QSAR predictions (Example4), and micro-dosing studies (Example 5)—rely on response data in orderto build a model. Accordingly, surrogates such as biomarkers andmetabolomics can be used as a means to obtain response data to build auseful model, particularly where the generation of response data isprohibitively expensive or time-consuming.

Example 7. Ex Vivo Testing and Personalized Medicine

Ex vivo testing results can be used to build the models for use with thesystems and methods taught herein. The term “ex vivo” can be used torefer to experimentation or measurements done in or on tissue in anenvironment outside the organism with minimum alteration of naturalconditions. Ex vivo conditions allow experimentation under morecontrolled conditions than is possible in in vivo experiments (in theintact organism), at the expense of altering the “natural” environment.A primary advantage of using ex vivo tissues is the ability to performtests or measurements that would otherwise not be possible or ethical inliving subjects. Examples of ex vivo testing would be studying thegrowth of bacteria in human cells and the associated antimicrobialactivity of potential antibiotics; or, studying the chemosensitivity offresh human hematopoietic cells, as well as malignant cells, in order toselect drugs with preferential toxicity to malignant cells.

As such, the results of ex vivo testing can be used to constructinput-response models of a particular subject and, based on that model,make predictions as to what types of therapeutic compounds might beeffective in yielding a desired response within that subject. Thesemodels would have to be able to capture the complex, nonlinear behaviorthat is present in cell-, tissue-, and organ-scale processes. Theability of the systems and methods taught herein to quickly provideaccurate and robust models of complex, nonlinear phenomena, asdemonstrated in the previous examples, makes them useful in theapplication of ex vivo testing. One of skill will appreciated thesignificant impact in the area of personalized medicine made possible bythe systems and methods taught herein; i.e., developing drug therapiesat a dosage that is most appropriate for an individual patient.

Example 8. Demand Forecasting

The systems and methods taught herein have many potential applicationsoutside of systems biology and drug design. For example, an importantarea of application is demand forecast modeling, where the input couldbe an individual consumer and the response is a product or service thatindividual might choose or require in the future. These products orservices could be, for example, retail consumer products, health careservices, or internet web sites.

In the case of QSAR modeling for biological applications, a model isbuilt using available data and the molecular properties of an inputcompound are mapped to the parameters of the model. This mapping is thenused to predict a certain response of interest based on the molecularproperties of the input. In the case of demand forecasting, a modelwould be built using available data on individuals and their observeddemand for products and services. The specific attributes of thoseindividuals could then be mapped to the parameters of the demand model.

Using demand forecasting for mapping, one could predict a future demandfor products and services based solely, for example, on one or morespecific attributes of an individual. This type of modeling, and thepredictions they would allow, would be very valuable for consumerproducts manufacturers, health care service providers, and those tryingto reach potential customers through online web services.

Example 9. Implementation of the Algorithms and Optimization of ResponseFunction Variables

This example shows the implementation of the algorithms and optimizationof response function variables for use in the systems and methods taughtherein.

9.1 Algorithm

Take the following steps:

-   1) Read in data: t_(i), f_(i); i=1, . . . , npts, where npts is the    total number of points in all the data sets;-   2) Normalize all data values: f_(i)*=f_(i)/scale, where:

${f\;{scale}} = \{ {\begin{matrix}{C_{0},} & {{if}\mspace{14mu}{response}\mspace{14mu}{species}{\mspace{11mu}\;}{is}\mspace{14mu}{the}\mspace{14mu}{same}\mspace{14mu}{as}\mspace{14mu}{input}\mspace{14mu}{species}} \\{1,} & {otherwise}\end{matrix};} $

-   3) Transform data:

${{\hat{t}}_{i} = \frac{t_{i}}{t_{\max}}},$

where t_(max) is the largest t_(i) value (smallest t_(i) value isassumed to be 0)

${{\hat{f}}_{i} = \frac{f_{i}^{*} + f_{\max}^{*} - {2f_{\min}^{*}}}{f_{\max}^{*} - f_{\min}^{*}}},$

where f_(min)* and f_(max)* are the smallest and largest f_(i)* values;

-   4) Fit data to the equation:

$\begin{matrix}{{\hat{M_{0}^{0}} + {\hat{M_{0}^{1}}({kernel})} + {\lbrack {\hat{M_{1}^{0}} + {\hat{M_{1}^{1}}({kernel})}} \rbrack\{ \frac{1 - e^{{\lbrack{\hat{N_{1}^{0}} + {\hat{N_{1}^{1}}{({kernel})}}}\rbrack}\hat{t}}}{1 + {( {e^{K} - 2} )e^{{\lbrack{\hat{N_{1}^{0}} + {\hat{N_{1}^{1}}{({kernel})}}}\rbrack}\hat{t}}}} \}} + \ldots + {\lbrack {\hat{M_{n}^{0}} + {\hat{M_{n}^{1}}({kernel})}} \rbrack\{ \frac{1 - e^{{\lbrack{\hat{N_{n}^{0}} + {\hat{N_{n}^{1}}{({kernel})}}}\rbrack}\hat{t}}}{1 + {( {e^{K} - 2} )e^{{\lbrack{\hat{N_{n}^{0}} + {\hat{N_{n}^{1}}{({kernel})}}}\rbrack}\hat{t}}}} \}}} = \hat{f}} & (27)\end{matrix}$

where kernel is defined as:

${{kernel} \equiv \frac{1 - e^{\alpha_{p}C_{0}}}{1 + {( {e^{K} - 2} )e^{{- \alpha_{p}}C_{0}}}}},$

andthis is done by minimizing the following objective function for K,K_(p), α_(p), M{circumflex over ( )}₀ ⁰, . . . , M{circumflex over( )}_(n) ⁰, M{circumflex over ( )}₀ ¹, . . . , M{circumflex over( )}_(n) ¹, N{circumflex over ( )}₁ ⁰, . . . , N{circumflex over( )}_(n) ⁰, and N{circumflex over ( )}₁ ¹, . . . , N{circumflex over( )}_(n) ¹ (see section 9.2 for details of the minimization procedure):

$\begin{matrix}{F = {\sum\limits_{i = 1}^{npts}\{ {\lbrack {\hat{M_{0}^{0}} + {\hat{M_{0}^{1}}({kernel})}} \rbrack + {\lbrack {\hat{M_{1}^{0}} + {\hat{M_{1}^{1}}({kernel})}} \rbrack\{ \frac{1 - e^{{\lbrack{\hat{N_{1}^{0}} + {\hat{N_{1}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}{1 + {( {e^{K} - 2} )e^{{\lbrack{\hat{N_{1}^{0}} + {\hat{N_{1}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}} \}} + \ldots +  \quad{{\lbrack {\hat{M_{n}^{0}} + {\hat{M_{n}^{1}}({kernel})}} \rbrack\{ \frac{1 - e^{{\lbrack{\hat{N_{n}^{0}} + {\hat{N_{n}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}{1 + {( {e^{K} - 2} )e^{{\lbrack{\hat{N_{n}^{0}} + {\hat{N_{n}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}} \}} = {\hat{f}}_{i}} \}^{2}} }} & (28)\end{matrix}$

-   5) Transform response function variables back to original space    (t_(i), f_(i)):

$\hat{M_{0}^{0}} = { \frac{M_{0}^{0} + f_{\max}^{*} - {2f_{\min}^{*}}}{f_{\max}^{*} - f_{\min}^{*}}\Leftrightarrow M_{0}^{0}  = {{\hat{M_{0}^{0}}( {f_{\max}^{*} - f_{\min}^{*}} )} - ( {f_{\max}^{*} - {2f_{\min}^{*}}} )}}$${{\hat{M_{j}^{0}} = { \frac{M_{j}^{0}}{f_{\max}^{*} - f_{\min}^{*}}\Leftrightarrow M_{j}^{0}  = {( {f_{\max}^{*} - f_{\min}^{*}} )\hat{M_{j}^{0}}}}};{j = 1}},\ldots\mspace{14mu},n$${{\hat{M_{j}^{0}} = { \frac{M_{j}^{1}}{f_{\max}^{*} - f_{\min}^{*}}\Leftrightarrow M_{j}^{1}  = {( {f_{\max}^{*} - f_{\min}^{*}} )\hat{M_{j}^{0}}}}};{j = 0}},\ldots\mspace{14mu},n$${{\hat{N_{j}^{0}} = { {N_{j}^{0}t_{\max}}\Leftrightarrow N_{j}^{0}  = \frac{\hat{N_{j}^{0}}}{t_{\max}}}};{j = 1}},\ldots\mspace{14mu},n$${{\hat{N_{j}^{1}} = { {N_{j}^{1}t_{\max}}\Leftrightarrow N_{j}^{1}  = \frac{\hat{N_{j}^{1}}}{t_{\max}}}};{j = 1}},\ldots\mspace{14mu},n$

This will yield a final response function in terms of time and initialdose,

$\begin{matrix}{{f(t)} = {{\lbrack {M_{0}^{0} + {M_{0}^{1}({kernel})}} \rbrack({fscale})} + {\quad{{{\lbrack {M_{1}^{0} + {M_{1}^{1}({kernel})}} \rbrack({fscale})\{ \frac{1 - e^{{\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}t}}{1 + {( {e^{K} - 2} )e^{{\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}t}}} \}} + \ldots\mspace{11mu} + {\lbrack {M_{n}^{0} + {M_{n}^{1}({kernel})}} \rbrack( {f\;{scale}} )\{ \frac{1 - e^{{\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}t}}{1 + {( {e^{K} - 2} )e^{{\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}t}}} \}}};}}}} & (29)\end{matrix}$

and,this function will serve as the prediction for a full time-courseresponse for a given dose.Transforming Equation 27 into Equation 29:

Substituting relationships from [3.] and [5.] into Equation (27) gives:

${\frac{M_{0}^{0} + f_{\max}^{*} - {2f_{\min}^{*}}}{f_{\max}^{*} - f_{\min}^{*}} + \frac{M_{0}^{1}({kernel})}{f_{\max}^{*} - f_{\min}^{*}} + {\lbrack {\frac{M_{1}^{0}}{f_{\max}^{*} - f_{\min}^{*}} + \frac{M_{1}^{1}({kernel})}{f_{\max}^{*} - f_{\min}^{*}}} \rbrack\{ \frac{1 - e^{{\lbrack{{N_{1}^{0}{(t_{\max})}} + {{N_{1}^{1}{(t_{\max})}}{({kernel})}}}\rbrack}\frac{t}{t_{\max}}}}{1 + {( {e^{K} - 2} )e^{{\lbrack{{N_{1}^{0}{(t_{\max})}} + {{N_{1}^{1}{(t_{\max})}}{({kernel})}}}\rbrack}\frac{t}{t_{\max}}}}} \}} + \ldots + {\lbrack {\frac{M_{n}^{0}}{f_{\max}^{*} - f_{\min}^{*}} + \frac{M_{n}^{1}({kernel})}{f_{\max}^{*} - f_{\min}^{*}}} \rbrack\{ \frac{1 - e^{{\lbrack{{N_{n}^{0}{(t_{\max})}} + {{N_{n}^{1}{(t_{\max})}}{({kernel})}}}\rbrack}\frac{t}{t_{\max}}}}{1 + {( {e^{K} - 2} )e^{{\lbrack{{N_{n}^{0}{(t_{\max})}} + {{N_{n}^{1}{(t_{\max})}}{({kernel})}}}\rbrack}\frac{t}{t_{\max}}}}} \}}} = \frac{f^{*} + f_{\max}^{*} - {2f_{\min}^{*}}}{f_{\max}^{*} - f_{\min}^{*}}$

Cancelling the f*_(max)−f*_(min) and t_(max) terms gives:

${{M_{0}^{0} + f_{\max}^{*} - {2f_{\min}^{*}} + {M_{0}^{1}({kernel})} + {\lbrack {M_{1}^{0} + {M_{1}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}t}}{1 + {( {e^{K} - 2} )e^{{\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}t}}} \}} + \ldots + {\lbrack {M_{n}^{0} + {M_{n}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}t}}{1 + {( {e^{K} - 2} )e^{{\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}t}}} \}}} = {f^{*} + f_{\max}^{*} - {2f_{\min}^{*}}}};$

The f*_(max) and 2f*_(min) terms cancel out, and f*=f/fscale, whichgives:

${M_{0}^{0} + {M_{0}^{1}({kernel})} + {\lbrack {M_{1}^{0} + {M_{1}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}t}}{1 + {( {e^{K} - 2} )e^{{\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}t}}} \}} + \ldots + {\lbrack {M_{n}^{0} + {M_{n}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}t}}{1 + {( {e^{K} - 2} )e^{{\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}t}}} \}}} = \frac{f}{fscale}$

Multiplying both sides by fscale gives:

${\lbrack {M_{0}^{0} + {M_{0}^{1}({kernel})}} \rbrack({fscale})} + {\quad{{{{\lbrack {M_{1}^{0} + {M_{1}^{1}({kernel})}} \rbrack({fscale})\{ \frac{1 - e^{{\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}t}}{1 + {( {e^{K} - 2} )e^{{\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}t}}} \}} + \ldots + {\lbrack {M_{n}^{0} + {M_{n}^{1}({kernel})}} \rbrack({fscale})\{ \frac{1 - e^{{\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}t}}{1 + {( {e^{K} - 2} )e^{{\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}t}}} \}}} = f};}}$

which is equivalent to Equation (29).

9.2 Optimization of Response Function Variables

The optimization procedure consists of a set of nested optimizations forthe response function variables K, K_(p), α_(p), and the N{circumflexover ( )}_(j) ⁰ and N{circumflex over ( )}_(j) ¹'s:

-   -   Perform a one-dimensional bounded search to find the K value        (note: there is only one K value across multiple data sets        within a given experiment) that minimizes a function whose value        is determined by        -   Performing a one-dimensional bounded search to find the            K_(p) value that minimizes a function whose value is            determined by            -   Performing a one-dimensional bounded search to find the                α_(p) value that minimizes a function whose value is                determined by                -   Cycling through a series of two-dimensional bounded,                    adaptive grid-refinement searches to find the                    N{circumflex over ( )}_(j) ⁰ and N{circumflex over                    ( )}_(j) ¹ values that minimize the objective                    function F, Equation (28)

To calculate M{circumflex over ( )}_(j) ⁰ and M{circumflex over ( )}_(j)¹'s,

-   -   (i) start with the objective function, Equation (28),

$F = {\sum\limits_{i = 1}^{npts}\{ {\lbrack {\hat{M_{0}^{0}} + {\hat{M_{0}^{1}}({kernel})}} \rbrack + {\quad{{\lbrack {\hat{M_{1}^{0}} + {\hat{M_{1}^{1}}({kernel})}} \rbrack\{ \frac{1 - e^{{\lbrack{\hat{N_{1}^{0}} + {\hat{N_{1}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}{1 + {( {e^{K} - 2} )e^{{\lbrack{\hat{N_{1}^{0}} + {\hat{N_{1}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}} \}} + \ldots +  \quad{{\lbrack {\hat{M_{n}^{0}} + {\hat{M_{n}^{1}}({kernel})}} \rbrack\{ \frac{1 - e^{{\lbrack{\hat{N_{n}^{0}} + {\hat{N_{n}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}{1 + {( {e^{K} - 2} )e^{{\lbrack{\hat{N_{n}^{0}} + {\hat{N_{n}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}} \}} = {\hat{f}}_{i}} \}^{2}}}} }$

and,

-   -   (ii) solve the system of 2(n+1) linear equations that results        from setting

${{\frac{\partial F}{\partial\hat{M_{j}^{0}}} = 0};{j = 0}},\ldots\mspace{14mu},{{{n\mspace{14mu}{and}\mspace{14mu}\frac{\partial F}{\partial\hat{M_{j}^{1}}}} = 0};{j = 0}},\ldots\mspace{14mu},n$$\frac{\partial F}{\partial\hat{M_{0}^{0}}} = {{2{\sum\limits_{i = 1}^{npts}{\{\}}}} = 0}$${\frac{\partial F}{\partial\hat{M_{0}^{1}}}2{\sum\limits_{i = 1}^{npts}{({kernel}){\{\}}}}} = 0$$\frac{\partial F}{\partial\hat{M_{1}^{0}}} = {{2{\sum\limits_{i = 1}^{npts}{{\{\}}\lbrack \frac{1 - e^{{\lbrack{\hat{N_{1}^{0}} + {\hat{N_{1}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}{1 + {( {e^{K} - 2} )e^{{\lbrack{\hat{N_{1}^{0}} + {\hat{N_{1}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}} \rbrack}}} = 0}$$\frac{\partial F}{\partial\hat{M_{1}^{1}}} = {{2{\sum\limits_{i = 1}^{npts}{({kernel}){{\{\}}\lbrack \frac{1 - e^{{\lbrack{\hat{N_{1}^{0}} + {\hat{N_{1}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}{1 + {( {e^{K} - 2} )e^{{\lbrack{\hat{N_{1}^{0}} + {\hat{N_{1}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}} \rbrack}}}} = 0}$⋮$\frac{\partial F}{\partial\hat{M_{n}^{0}}} = {{2{\sum\limits_{i = 1}^{npts}{{\{\}}\lbrack \frac{1 - e^{{\lbrack{\hat{N_{n}^{0}} + {\hat{N_{n}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}{1 + {( {e^{K} - 2} )e^{{\lbrack{\hat{N_{n}^{0}} + {\hat{N_{n}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}} \rbrack}}} = 0}$${\frac{\partial F}{\partial\hat{M_{n}^{1}}} = {{2{\sum\limits_{i = 1}^{npts}{({kernel}){{\{\}}\lbrack \frac{1 - e^{{\lbrack{\hat{N_{n}^{0}} + {\hat{N_{n}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}{1 + {( {e^{K} - 2} )e^{{\lbrack{\hat{N_{n}^{0}} + {\hat{N_{n}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}} \rbrack}}}} = 0}};$${where},{{\{\}} = \{ {\lbrack {\hat{M_{0}^{0}} + {\hat{M_{0}^{1}}({kernel})}} \rbrack + {\quad{{\lbrack {\hat{M_{1}^{0}} + {\hat{M_{1}^{1}}({kernel})}} \rbrack\{ \frac{1 - e^{{\lbrack{\hat{N_{1}^{0}} + {\hat{N_{1}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}{1 + {( {e^{K} - 2} )e^{{\lbrack{\hat{N_{1}^{0}} + {\hat{N_{1}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}} \}} + \ldots +  \quad{{\lbrack {\hat{M_{n}^{0}} + {\hat{M_{n}^{1}}({kernel})}} \rbrack\{ \frac{1 - e^{{\lbrack{\hat{N_{n}^{0}} + {\hat{N_{n}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}{1 + {( {e^{K} - 2} )e^{{\lbrack{\hat{N_{n}^{0}} + {\hat{N_{n}^{1}}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}} \}} = {\hat{f}}_{i}} \}}}} }$

Rearranging yields:

${{{\overset{\sim}{M}}_{0}^{0}{\sum\limits_{i = 1}^{npts}\; 1}} + {{\hat{M}}_{0}^{1}{\sum\limits_{i = 1}^{npts}\; 0}} + {{\hat{M}}_{1}^{0}{\sum\limits_{i = 1}^{npts}\;\lbrack 1\rbrack}} + {{\hat{M}}_{1}^{1}{\sum\limits_{i = 1}^{npts}\;{0\lbrack 1\rbrack}}} + \ldots\mspace{14mu} + {{\hat{M}}_{n}^{0}{\sum\limits_{i = 1}^{npts}\;\lbrack n\rbrack}} + {{\hat{M}}_{n}^{1}{0\lbrack n\rbrack}}} = {\sum\limits_{i = 1}^{npts}\;{\overset{\sim}{f}}_{i}}$${{{\hat{M}}_{0}^{0}{\sum\limits_{i = 1}^{npts}\; 0}} + {{\hat{M}}_{0}^{1}{\sum\limits_{i = 1}^{npts}\; 0^{2}}} + {{\hat{M}}_{1}^{0}{\sum\limits_{i = 1}^{npts}\;{0\lbrack 1\rbrack}}} + {{\hat{M}}_{1}^{1}{\sum\limits_{i = 1}^{npts}\;{0^{2}\lbrack 1\rbrack}}} + \ldots\mspace{14mu} + {{\hat{M}}_{n}^{0}{\sum\limits_{i = 1}^{npts}\;{0\lbrack n\rbrack}}} + {{\hat{M}}_{n}^{1}{\sum\limits_{i = 1}^{npts}\;{0^{2}\lbrack n\rbrack}}}} = {\sum\limits_{i = 1}^{npts}\;{0{\hat{f}}_{i}}}$${{{\overset{\sim}{M}}_{0}^{0}{\sum\limits_{i = 1}^{npts}\;\lbrack 1\rbrack}} + {{\overset{\sim}{M}}_{0}^{1}{\sum\limits_{i = 1}^{npts}\;{0\lbrack 1\rbrack}}} + {{\hat{M}}_{1}^{0}{\sum\limits_{i = 1}^{npts}\;\lbrack 1\rbrack^{2}}} + {{\hat{M}}_{1}^{1}{\sum\limits_{i = 1}^{npts}\;{0\lbrack 1\rbrack}^{2}}} + \ldots\mspace{14mu} + {{\hat{M}}_{n}^{0}{\sum\limits_{i = 1}^{npts}\;{\lbrack 1\rbrack\lbrack n\rbrack}}} + {{\hat{M}}_{n}^{1}{\sum\limits_{i = 1}^{npts}\;{{0\lbrack 1\rbrack}\lbrack n\rbrack}}}} = {\sum\limits_{i = 1}^{npts}\;{\lbrack 1\rbrack{\hat{f}}_{i}}}$${{{\hat{M}}_{0}^{0}{\sum\limits_{i = 1}^{npts}\;{0\lbrack 1\rbrack}}} + {{\overset{\sim}{M}}_{0}^{1}{\sum\limits_{i = 1}^{npts}\;{0^{2}\lbrack 1\rbrack}}} + {{\overset{\sim}{M}}_{1}^{0}{\sum\limits_{i = 1}^{npts}\;{0\lbrack 1\rbrack}^{2}}} + {{\hat{M}}_{1}^{1}{\sum\limits_{i = 1}^{npts}\;{0^{2}\lbrack 1\rbrack}^{2}}} + \ldots\mspace{14mu} + {{\hat{M}}_{n}^{0}{\sum\limits_{i = 1}^{npts}\;{{0\lbrack 1\rbrack}\lbrack n\rbrack}}} + {{\hat{M}}_{n}^{1}{\sum\limits_{i = 1}^{npts}\;{{0^{2}\lbrack 1\rbrack}\lbrack n\rbrack}}}} = {\sum\limits_{i = 1}^{npts}\;{{0\lbrack 1\rbrack}{\hat{f}}_{i}}}$⋮${{{\hat{M}}_{0}^{0}{\sum\limits_{i = 1}^{npts}\;\lbrack n\rbrack}} + {{\hat{M}}_{0}^{1}{\sum\limits_{i = 1}^{npts}\;{0\lbrack n\rbrack}}} + {{\hat{M}}_{1}^{0}{\sum\limits_{i = 1}^{npts}\;{\lbrack 1\rbrack\lbrack n\rbrack}}} + {{\hat{M}}_{1}^{1}{\sum\limits_{i = 1}^{npts}\;{{0\lbrack 1\rbrack}\lbrack n\rbrack}}} + \ldots\mspace{14mu} + {{\hat{M}}_{0}^{0}{\sum\limits_{i = 1}^{npts}\;\lbrack n\rbrack^{2}}} + {{\hat{M}}_{n}^{1}{\sum\limits_{i = 1}^{npts}\;{0\lbrack n\rbrack}^{2}}}} = {\sum\limits_{i = 1}^{npts}\;{\lbrack n\rbrack{\hat{f}}_{i}}}$${{{\hat{M}}_{0}^{0}{\sum\limits_{i = 1}^{npts}\;{0\lbrack n\rbrack}}} + {{\hat{M}}_{0}^{1}{\sum\limits_{i = 1}^{npts}\;{0^{2}\lbrack n\rbrack}}} + {{\hat{M}}_{1}^{0}{\sum\limits_{i = 1}^{npts}\;{{0\lbrack 1\rbrack}\lbrack n\rbrack}}} + {{\hat{M}}_{1}^{1}{\sum\limits_{i = 1}^{npts}\;{{0^{2}\lbrack 1\rbrack}\lbrack n\rbrack}}} + \ldots\mspace{14mu} + {{\hat{M}}_{0}^{0}{\sum\limits_{i = 1}^{npts}\;{0\lbrack n\rbrack}^{2}}} + {{\hat{M}}_{n}^{1}{\sum\limits_{i = 1}^{npts}\;{0^{2}\lbrack n\rbrack}^{2}}}} = {\sum\limits_{i = 1}^{npts}\;{{0\lbrack n\rbrack}{\hat{f}}_{i}}}$${where},{0 = {{({kernel})\mspace{14mu}{{and}\mspace{14mu}\lbrack j\rbrack}} = \lbrack \frac{1 - e^{{\lbrack{{\hat{N}}_{j}^{0} + {{\hat{N}}_{j}^{1}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}{1 + {( {e^{k} - 2} )e^{{\lbrack{{\hat{N}}_{j}^{0} + {{\hat{N}}_{j}^{1}{({kernel})}}}\rbrack}{\hat{t}}_{i}}}} \rbrack}},\mspace{14mu}{j = 1},\ldots\mspace{14mu},n$

This yields the following system of equations, in matrix form:

${\quad{\begin{bmatrix}{\sum\; 1} & {\sum 0} & {\sum\lbrack 1\rbrack} & {\sum{0\lbrack 1\rbrack}} & \ldots & {\sum\lbrack n\rbrack} & {\sum{0\lbrack n\rbrack}} \\{\sum 0} & {\sum 0^{2}} & {\sum{0\lbrack 1\rbrack}} & {\sum{0^{2}\lbrack 1\rbrack}} & \ldots & {\sum{0\lbrack n\rbrack}} & {\sum{0^{2}\lbrack n\rbrack}} \\{\sum\lbrack 1\rbrack} & {\sum{0\lbrack 1\rbrack}} & {\sum\lbrack 1\rbrack^{2}} & {\sum{0\lbrack 1\rbrack}^{2}} & \ldots & {\sum{\lbrack 1\rbrack\lbrack n\rbrack}} & {\sum{{0\lbrack 1\rbrack}\lbrack n\rbrack}} \\{\sum{0\lbrack 1\rbrack}} & {\sum{0^{2}\lbrack 1\rbrack}} & {\sum{0\lbrack 1\rbrack}^{2}} & {\sum{0^{2}\lbrack 1\rbrack}^{2}} & \ldots & {\sum{{0\lbrack 1\rbrack}\lbrack n\rbrack}} & {\sum{{0^{2}\lbrack 1\rbrack}\lbrack n\rbrack}} \\\vdots & \vdots & \vdots & \vdots & \vdots & \vdots & \vdots \\{\sum\lbrack n\rbrack} & {\sum{0\lbrack n\rbrack}} & {\sum{\lbrack 1\rbrack\lbrack n\rbrack}} & {\sum{{0\lbrack 1\rbrack}\lbrack n\rbrack}} & \ldots & {\sum\lbrack n\rbrack^{2}} & {\sum{0\lbrack n\rbrack}^{2}} \\{\sum{0\lbrack n\rbrack}} & {\sum{0^{2}\lbrack n\rbrack}} & {\sum{{0\lbrack 1\rbrack}\lbrack n\rbrack}} & {\sum{{0^{2}\lbrack 1\rbrack}\lbrack n\rbrack}} & \ldots & {\sum{0\lbrack n\rbrack}^{2}} & {\sum{0^{2}\lbrack n\rbrack}^{2}}\end{bmatrix}\lbrack \begin{matrix}{\hat{M}}_{0}^{0} \\{\hat{M}}_{0}^{1} \\{\hat{M}}_{1}^{0} \\{\hat{M}}_{1}^{1} \\\vdots \\{\hat{M}}_{n}^{0} \\{\hat{M}}_{n}^{1}\end{matrix} \rbrack}} = {\quad\begin{bmatrix}{\sum{\hat{f}}_{i}} \\{\sum{0{\hat{f}}_{i}}} \\{\sum{\lbrack 1\rbrack{\hat{f}}_{i}}} \\{\sum{{0\lbrack 1\rbrack}{\hat{f}}_{i}}} \\\vdots \\{\sum{\lbrack n\rbrack{\hat{f}}_{i}}} \\{\sum{{0\lbrack n\rbrack}{\hat{f}}_{i}}}\end{bmatrix}}$

When a solution to this system of equations is required, the C₀, K,K_(p), α_(p), N{circumflex over ( )}_(j) ⁰ and N{circumflex over( )}_(j) ¹ values are known. Therefore, the C₀, K_(p), and α_(p) valuesare used to calculate a (kernel) value, and the (kernel), K,N{circumflex over ( )}_(j) ⁰ and N{circumflex over ( )}_(j) ¹ values areused to calculate the [j] values, (j=1, . . . , n). Given a (kernel)value and the [j] values, the linear system of equations (shown above)can be solved to give all of the M{circumflex over ( )}_(j) ⁰ andM{circumflex over ( )}_(j) ¹ values (j=0, . . . , n).

We claim:
 1. An unambiguous method of predicting a non-linear,time-dependent response of a component of a system to an input into thesystem, the method comprising: mapping input properties to modelparameters, the mapping including developing one-to-one relationshipsbetween model parameters and model output to obtain a specificity ofresults between model parameters and model output to get a uniqueinput-response relationship, the mapping including creating a model byidentifying the system, the component, the input, and the non-linear,time-dependent response; wherein, the input includes a set of actualinputs and a test input, and the non-linear time-dependent responseincludes a set of non-linear, time-dependent actual responses and anon-linear, test response; obtaining the set of non-linear,time-dependent actual responses of the component to the set of actualinputs; using the set of actual inputs and the set of non-linear,time-dependent actual responses to provide a model for predicting thenon-linear, test response to the test input, the model comprising theformula${C(t)} = {\lbrack {M_{0}^{0} + {M_{0}^{1}({kernel})}} \rbrack + {\quad{{\lbrack {M_{1}^{0} + {M_{1}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}{1 + {( {e^{K} - 2} )e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}} \}} + \ldots\mspace{14mu} + {\lbrack {M_{n}^{0} + {M_{n}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}{1 + {( {e^{K} - 2} )e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}} \}}}}}$wherein, M⁰ ₀, . . . , M⁰ _(n) and M¹ ₀, . . . , M¹ _(n) are overallscaling parameters; N⁰ ₁, . . . , N⁰ _(n) and N¹ ₁, . . . , N¹ _(n) areexponential scaling parameters; n ranges from 1 to 4; K is an overallshifting parameter; and, C(t) is the non-linear, time-dependent responseto the test input at time t; and,${{kernel} \equiv \frac{1 - e^{\alpha_{p}C_{0}}}{1 + {( {e^{K_{p}} - 2} )e^{{- \alpha_{p}}C_{0}}}}};$wherein, C₀ is the initial amount of the test input; K_(p) is a shiftingparameter related to C₀; and, α_(p) is shifting and scaling parameterrelated to C₀; and, using the model in the mapping to obtain thenon-linear, time-dependent test response to the test input; wherein, themapping provides the unambiguous prediction of the non-linear,time-dependent response of the component of a system to the input intothe system.
 2. The method of claim 1, wherein the system is anenvironmental system and the component is selected from the groupconsisting of air, water, and soil.
 3. The method of claim 1, whereinthe system is a mammal, and the component is selected from the groupconsisting of a cell, a tissue, an organ, a DNA, a virus, a protein, anantibody, a bacteria.
 4. The method of claim 1, wherein the system is achemical system.
 5. The method of claim 1, wherein the system is amechanical system.
 6. The method of claim 1, wherein the system is anelectrical system.
 7. An unambiguous method of predicting a non-linear,time-dependent response of a component of a mammalian system to an inputinto the system, the method comprising: mapping input properties tomodel parameters, the mapping including developing one-to-onerelationships between model parameters and model output to obtain aspecificity of results between model parameters and model output to geta unique input-response relationship, the mapping including creating amodel by selecting a component of the system, the component selectedfrom the group consisting of a cell, a tissue, an organ, a DNA, a virus,a protein, an antibody, a bacteria; selecting a set of actual inputs,the set of actual inputs having an element selected from the groupconsisting of a DNA, a virus, a protein, an antibody, a bacteria, achemical, a dietary supplement, a nutrient, and a drug; obtaining a setof non-linear, time-dependent actual responses of the component to theset of actual inputs; using the set of actual inputs and the set ofnon-linear, time-dependent actual responses to provide a model forpredicting a non-linear, test response to a test input, the modelcomprising the formula $\begin{matrix}{{{{{{C(t)} = {\lbrack {M_{0}^{0} + {M_{0}^{1}({kernel})}} \rbrack +}}\quad}\quad}\quad}{\quad{{\lbrack {M_{1}^{0} + {M_{1}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}{1 + {( {e^{K} - 2} )e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}} \}} + \ldots\mspace{14mu} + {\lbrack {M_{n}^{0} + {M_{n}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}{1 + {( {e^{K} - 2} )e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}} \}}}}} & (9)\end{matrix}$ wherein, M⁰ ₀, . . . , M⁰ _(n) and M¹ ₀, . . . , M¹ _(n)are overall scaling parameters; N⁰ ₁, . . . , N⁰ _(n) and N¹ ₁, . . . ,N¹ _(n) are exponential scaling parameters; n ranges from 1 to 4; K isan overall shifting parameter; and, C(t) is the non-linear,time-dependent response to the test input at time t; and,${{kernel} \equiv \frac{1 - e^{\alpha_{p}C_{0}}}{1 + {( {e^{K_{p}} - 2} )e^{{- \alpha_{p}}C_{0}}}}};$wherein, C₀ is the initial amount of the test input; K_(p) is a shiftingparameter related to C₀; and, α_(p) is shifting and scaling parameterrelated to C₀; and, using the model to obtain the non-linear,time-dependent test response to the test input wherein, the mappingprovides the unambiguous prediction of the non-linear, time-dependentresponse of the component of a system to the input into the system. 8.The method of claim 7, wherein the component is blood.
 9. The method ofclaim 7, wherein the component is a tumor cell.
 10. The method of claim7, wherein the component is a virus.
 11. The method of claim 7, whereinthe component is a bacteria.
 12. The method of claim 7, wherein thenon-linear, test response is a bacterial load.
 13. The method of claim7, wherein the non-linear, test response is a viral load.
 14. The methodof claim 7, wherein the non-linear, test response is a tumor marker. 15.The method of claim 7, wherein the non-linear, test response is a bloodchemistry.
 16. The method of claim 7, wherein the set of actual inputsincludes a set of dosages of a drug.
 17. The method of claim 7, whereinthe set of actual inputs includes a set of drugs.
 18. The method ofclaim 7, wherein the input is a diabetes drug, and the non-linear,time-dependent response is glucose in the bloodstream.
 19. A device forunambiguously predicting a non-linear, time-dependent response of acomponent of a physical system to an input into the system, the devicecomprising: a processor; a database for storing a set of actual inputdata, a set of non-linear, time-dependent actual response data, testinput data, and non-linear, time-dependent test response data on anon-transitory computer readable medium; the database containing dataused for establishing one-to-one relationships between model parametersand model output to obtain a specificity of results between modelparameters and model output to get a unique input-response relationship;an enumeration engine on a non-transitory computer readable medium toparameterize a non-compartmental model for unambiguously predicting anon-linear, test response to a test input, the non-compartmental modelcomprising the formula $\begin{matrix}{{{{C(t)} = {\lbrack {M_{0}^{0} + {M_{0}^{1}({kernel})}} \rbrack +}}\quad}{\quad{{\lbrack {M_{1}^{0} + {M_{1}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}{1 + {( {e^{K} - 2} )e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}} \}} + \ldots\mspace{14mu} + {\lbrack {M_{n}^{0} + {M_{n}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}{1 + {( {e^{K} - 2} )e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}} \}}}}} & (9)\end{matrix}$ wherein, M⁰ ₀, . . . , M⁰ _(n) and M¹ ₀, . . . , M¹ _(n)are overall scaling parameters; N⁰ ₁, . . . , N⁰ _(n) and N¹ ₁, . . . ,N¹ _(n) are exponential scaling parameters; n ranges from 1 to 4; K isan overall shifting parameter; and, C(t) is the non-linear,time-dependent response to the test input at time t; and,${{kernel} \equiv \frac{1 - e^{\alpha_{p}C_{0}}}{1 + {( {e^{K_{p}} - 2} )e^{{- \alpha_{p}}C_{0}}}}};$wherein, C₀ is the initial amount of the test input; K_(p) is a shiftingparameter related to C₀; and, α_(p) is shifting and scaling parameterrelated to C₀; and, a transformation module on a non-transitory computerreadable medium to transform the test data into the non-linear,time-dependent response data using the non-compartmental model, thetransformation module providing the one-to-one relationships betweenmodel parameters and model output to obtain the specificity of resultsbetween model parameters and model output to get the uniqueinput-response relationship.
 20. The device of claim 19, wherein thesystem is an environmental system and the component is selected from thegroup consisting of air, water, and soil.
 21. A device for unambiguouslypredicting a non-linear, time-dependent response of a component of amammalian system to an input into the system, the device comprising: aprocessor; a database for storing a set of actual input data, a set ofnon-linear, time-dependent actual response data, test input data, andnon-linear, time-dependent test response data on a non-transitorycomputer readable medium; the database containing data used forestablishing one-to-one relationships between model parameters and modeloutput to obtain a specificity of results between model parameters andmodel output to get a unique input-response relationship; an enumerationengine on a non-transitory computer readable medium to parameterize anon-compartmental model for unambiguously predicting a non-linear, testresponse to a test input, the non-compartmental model comprising theformula $\begin{matrix}{{{{C(t)} = {\lbrack {M_{0}^{0} + {M_{0}^{1}({kernel})}} \rbrack +}}\quad}{\quad{{\lbrack {M_{1}^{0} + {M_{1}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}{1 + {( {e^{K} - 2} )e^{{- {\lbrack{N_{1}^{0} + {N_{1}^{1}{({kernel})}}}\rbrack}}t}}} \}} + \ldots\mspace{14mu} + {\lbrack {M_{n}^{0} + {M_{n}^{1}({kernel})}} \rbrack\{ \frac{1 - e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}{1 + {( {e^{K} - 2} )e^{{- {\lbrack{N_{n}^{0} + {N_{n}^{1}{({kernel})}}}\rbrack}}t}}} \}}}}} & (9)\end{matrix}$ wherein, M⁰ ₀, . . . , M⁰ _(n) and M¹ ₀, . . . , M¹ _(n)are overall scaling parameters; N⁰ ₁, . . . , N⁰ _(n) and N¹ ₁, . . . ,N¹ _(n) are exponential scaling parameters; n ranges from 1 to 4; K isan overall shifting parameter; and, C(t) is the non-linear,time-dependent response to the test input at time t; and,${{kernel} \equiv \frac{1 - e^{{- \alpha_{p}}C_{0}}}{1 + {( {e^{K_{p}} - 2} )e^{{- \alpha_{p}}C_{0}}}}};$wherein, C₀ is the initial amount of the test input; K_(p) is a shiftingparameter related to C₀; and, α_(p) is shifting and scaling parameterrelated to C₀; and, a transformation module on a non-transitory computerreadable medium to transform the test data into the non-linear,time-dependent response data using the non-compartmental model, thetransformation module providing the one-to-one relationships betweenmodel parameters and model output to obtain the specificity of resultsbetween model parameters and model output to get the uniqueinput-response relationship;
 22. The device of claim 21, wherein thecomponent is blood.
 23. The device of claim 21, wherein the component isa tumor cell.
 24. The device of claim 21, wherein the component is avirus.
 25. The device of claim 21, wherein the component is a bacteria.26. The device of claim 21, wherein the non-linear, time-dependentresponse is a bacterial load.
 27. The device of claim 21, wherein thenon-linear, time-dependent response is a viral load.
 28. The device ofclaim 21, wherein the non-linear, time-dependent response is a tumormarker.
 29. The device of claim 21, wherein the non-linear,time-dependent response is a blood chemistry.
 30. The device of claim21, wherein the device is a handheld device.